(from https://hinative.com/ja/questions/605857)
考えていくうちに→
As I think about it (further)

少しずつ次の状態になること(= gradually)を意味します。(考えていく has the meaning of 'gradual progression')
「考えると、」はちょっとの間考えただけで次の状態になり、「考えていくうちに、」はそれに比べてもっと長い時間考えてから、次の状態になることを意味します。
考えると represents that just by thinking (about it) for a short time, the next state is reached
考えていくうちに represents that after thinking (about it) for a longer time, the next state is reached
*ちょっとの間考えただけで恐ろしい。
 Just the thought of it is horrifying.
 (lit.)I just thought (about it), and it's scary

So 考えていくうちに can interpreted as 'While I think and think and think about it'

More Ex:
考えていくうちに、私は間違いに気付いた
As I thought and thought and thought about it,  I noticed my mistakes
As I thought about it further,  I noticed my mistakes
そうやって歩いているうちに、もう家に着いた
While walking like that, I arrived home
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子どもの頃から使っていくうちに、身についてきたものだからです。
(The て form) is something we master after using it again and again from childhood.
While/as you use and use (The て form) since childhood, it's something you have mastered (since that time until now).
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使っていくうちに雪化粧した富士山が現れる「エアイン 富士山消しゴム」(https://www.youtube.com/watch?v=nRhOtLdUWts_
As you use and use (the eraser), Mt. Fuji will appear