In the history of video game development, one would be hard-pressed to find a studio more prolific than Square-Enix. For the purposes of this paper, I will focus more exclusively on Square's titles before the merger of Square Enix. This is partially due to the fact that before the merger, Enix was markedly a publisher first and game development studio second, but I also want to focus on Square's most notable and successful series to illustrate the advancements in animation they have made while reducing other variables in order to more clearly draw how their animation style and method has changed over time.
In 1987, Square released Final Fantasy, which was developed bya team of 9 people. Amidst financial ruin looming over the company's future, the game could potentially have been their last title. The game starts with Nobuo Uematsu's famous arpeggio that has since found a place in many other titles in the series. You choose a name and begin on your adventure.
By today's standards, Final Fantasy's animations are rudimentary. And even for the time, they were not entirely unique, though they were an improvement on rival company Enix's Dragon quest in that the main sprites used to move around the maps could face different directions. They still, however, had only two frames per cycle.
Seperate and more detailed sprites were used when the player encountered an enemy and was transported to the battle screen. Now, instead of having a simple sprite with a total of 8 possible frames (2 per direction,) a sprite not only had more animation frames, but also had more detail and was significantly larger. This theme stuck with the series for over a decade and has been a defining trait for the series' animation.
The first three games in the series followed roughly the same formula, with little improvement in actual animation. This was largely due to hardware limitations of the famicom system. Final fantasy 4, or final fantasy II as it was released in the U.S., was the first title released on the super famicom or SNES (respectively) and due to the better hardware, much more could be accomplished.
Final fantasy II starts of with a cutscene where a series of animations tells the story, departing from the previous titles' use of text to illustrate the setting. This was well approached for such an early attempt at cutscene usage in an RPG, and feels almost choreographed, although someone playing back on the game today may fault the strange walking-in-place effect that plagued early 90's RPGs so thoroughly.
FFII still kept a 2 frame per direction animation, but added abilities with unique animations that would draw the sprite moving around the screen. Beyond this, not much more was accomplished toward the animations. The important development FFII accomplished was to introduce structured storytelling by moving sprites in pre-defined patterns while dialogue occured.
Final Fantasy V embraced more of the new technology available, allowing for more sprite states to be allowed during dialogue and cutscenes involving the player character. This minor addition of sprite states allowed for exponentially more clarity when trying to convey a character's intention, mood and reactions. This led to the game feeling much more alive; the characters could hold their hands up to a fire, shake their head, bow. Their eyes could grow comically when surprised and enunciated motions and proportions would help exaggerate emotion and give players a better feeling of presence in the story. The game also experimented with 3d rendering and scaling graphics during cutscenes.
Still limited by hardware, specifically the cartridge system that nintendo would not give up for another generation of consoles, they had reached their limit. Cartridges for the new game console Nintendo was planning on pushing out were capable of holding only 64 megabytes of data (http://www.videogameconsolelibrary.com/pg90-n64.htm#page=specs), whereas the new Sony Playstation used CDs, which were capable of holding 750 megabytes. Not only could the playstation hold more content per media type, but it would also allow a player to switch discs while playing, allowing essentially limitless amounts of content.
Final fantasy 7, arguably the most loved game in the series and certainly the best selling with 11 million units sold, was released in 1997 for the Playstation after having been developed for 3 years prior. Making full use of the 120 people involved in the game and a budget larger than 30 million dollars (https://web.archive.org/web/20071028163434/http://www.gamespot.com/features/vgs/universal/finalfantasy_hs/sec1_7_2.html), the game featured 3d graphics, often overlayed on top of painstakingly detailed backgrounds, but still retained certain traits in its animation style from previous games in the series - most notably a distinct disparity between "overworld" character models and in battle character models. A low poly character, made from colored basic shapes suddenly became a fully fledged character model when entering a battle.
This disparity was taken another level further, however, as cutscenes were now able to be pre rendered and loaded onto the discs to be played when an event triggered them. The detail in these cutscenes were as fleshed out as Square was willing to make them; the only requirement being time dedicated by a computer to render the scenes and time spent animating and modeling the characters and environment. Often, these scenes were so much more detailed than the rest of the game that they stand out significantly. Other times, the basic low poly character models were still used while being overlayed on top of pre-renderd graphics. This meant that you could, on occasion, control them while the cutscenes were playing.
This is one of the first examples of animation in games not being limited by hardware. Although these high quality cutscenes were pre-rendered, it was still important and necessary to convey a character correctly through animation and the amount of detail allowed suddenly opened up a world of options for that. Sometimes a limited palette of choices per animation - a la sprite animation circa Final Fantasy 5 and 6 - would give you a smoother creative flow. You had certain images that would display briefly and could cover a fairly wide range of situations. The dichotomy between these styles is that now if you wanted to create a fully fledged cutscene with multiple characters with the goal of fully exploiting a scene, you would have to animate every part of a character.  Composition, line of action, motion, sense of gravity, being true to character - all of these things were essentially born around this point in gaming, albeit on a less frequent occurence.  This essentially foreshadowed the direction game animation was heading.
There are pioneering times and there are innovative times.  This was a pioneering age for animation in gaming, and during this time, not everything is going to reflect well in retrospect. The experimentation in techniques and methodology are still being developed, and if you look back at the CGI cutscenes from Final Fantasy 7, you could point out a lot of flaws in their animation in contrast to today's standards.  Characters often seem weightless, sometimes it's a product of attempting to stylize a scene and somewhat forgivable, but it often just creates an effect of unrealistic toyishness.
However, this was new at the time, and for that reason, these weren't seen as flaws, because no other game animation had set a frame of reference beyond the scope of what was being accomplished.  This was essentially new territory save for a few other examples and the effect it had on people playing the games was fairly astounding.  Characters were more tangible than they had ever been, even if only for a brief moment.  Hair folicles moved independently, magic flowed smoothly and effects were on par with movie special effects.  It brought a much more cinematic feel to the game and helped flesh out the characters.  I assert that this is a core reason that the original playerbase from final fantasy 7 is so fiercely nostalgic about it.  In a way, these were the first video game characters that they got to meet, alongside getting to know them during an extremely lengthy campaign.
One character design point I would like to make, albeit a slight digression, is that the design for the characters is crafted from the ground up to make characters legible between all three stages that they are rendered in.  Every character is exaggerated so that even at the lowest poly count, they are immediately recognizable.  The protagonist, Cloud, has absurd hair that spikes out chaotically, Tifa has enormous breasts and shows a lot of skin, Barrett has a gun for an arm, is the only black character in the game and is at least twice the size of other characters, Aerith has a giant pink bow and bangs that look like wings and Sephiroth has similar hair.  This is a clever way to make your characters read as the same, regardless of the quality they are rendered at.
At the same time they were working on Final Fantasy 7, some of the staff working on the game were also working on Square's Chrono Trigger.  This game deserves an honorable mention tangent due largely to its unprecedented sprite animation.  The game features more than three thousand sprites, many of which are animated extensively.   (http://www.videogamesprites.net/ChronoTrigger/).
After development on Final Fantasy 7 resumed in 1995, many CG artists were brought on from hollywood CG studios (https://archive.org/stream/Computer_and_Video_Games_Issue_185_1997-04_EMAP_Images_GB#page/n87/mode/2up) to work in tandem with the developers in creating a unique cinematic experience through innovative use of CGI pre-rendered cutscenes.  Co-operation between japanese and western studios before had been fairly limited, especially in game development.  Final fantasy 7 was truly a turning point for the industry as a whole.
After pioneering, however, comes innovation and refinement.  Final Fantasy 8 saw a huge increase in fidelity regarding character models and animation, both in game and in the cutscenes.  The characters felt far more fleshed out and true to their characters without being too charicatured in order to maintain recognition between a low poly character and their battle model.  In fact, the models on the regular movement map were the same as the battle models, departing from a long standing tradition in Final Fantasy games.
This, however, is a point where we see the trend of overanimated characters in square's work.  I specifically refer to Zell here, the character with the faux-hawx and mike tyson-esque face tattoo who is introduced in a cgi scene wherein he is practicing kung-fu in a school hallway, looks directly at the camera, cartwheels towards it and poses with his thumb up.  While the series as a whole makes huge strides towards refining and perfecting CGI cutscenes and character development, the characters themselves begin to feel a little flat.
Final fantasy 9 was much more stylistic and fantastical than previous games, though it didn't necissarily suffer for that.  It captures that pretty well at some points, but the two main characters feel almost devoid of originality or any real enthusiasm behind their design and animation.  It became safe bet character development.  Whereas earlier games were uniquely thought out and characters felt original and interesting, 9 played everything two dimensionally safe.  The protagonist was a thief with a mysterious past after the love of a princess who wanted to rebel by running away from home and cutting her hair.
The most interesting character from 9, was a side character: Vivi, a black mage who turned out to be a construct meant to become a powerful destructive weapon.  His character design is deceptively simple, essentially he's some clothes and a hat with two yellow eyes poking out from the darkness under his hat.  The beautiful thing about Vivi as a character is he was a good example of subtlety in character animation.  Even the slightest turn of his head upwards made him feel extremely contemplative as he learned that he might die within a year.  This was the most refreshing part about Final Fantasy 9 becuase it was a contrast to the strange mixed trends of overanimated characters and flat, safe tropes as main characters.
From Final Fantasy 10 on out, there is really just a two-sided trend.  To oversimplify: Graphical fidelity, particularly concerning pre-rendered cutscenes improved greatly while character development and communication of character through animation stagnated.  Final Fantasy 10 takes place all around water.  The main character plays a sport underwater, basically every map has water and the antagonist creature in the game lives in the water.  Given all this, it's strange that it seems like the people animating the protagonist's character model have never swam in their entire lives.  None of the animations look good outside a few cutscenes, the characters are all poorly imagined and move in a stiff, uncomfortable fashion.
Stylistically, 12 was a great game.  The setting was great, the art style felt unique and fresh and the story had a lot of promise.  The animation was, on occasion, well done, but often characters would gesticulate well toward someone in a cutscene only to have the other character stand with their arms at their side, motionless and move only their head when they talked.  None of the main characters really felt individual, instead blending into each other.  Again, the pre-rendered CGI had gotten much, much better since the last game as was the trend.
Final fantasy 13 had very well animated characters that read as though they were who their character development would dictate, but the problem is that most of that character development is lost in a schizophrenically paced story filled with what felt like words made from random letters and glottal stops.  The focus on character animation was strong and well crafted, characters felt more real than in most final fantasy games, but there was one reason that this game suffered greatly: The game wasn't fun.  This series would have made a great movie but fell flat as a game.  The pre-rendered scenes were absolutely stunning and well crafted but being dropped back into control of the characters felt unwanted.
This is the dilemma: The trend of pre-rendered cutscenes started out as being ancillary to the game, and that slowly reversed to the point where the game now feels secondary to the cutscenes.  There's a lack of macro-subtlety in this approach but it could also be considered an interactive movie of sorts, so it really is a matter of perspective as to whether or not this is a bad thing.