Moving between servers best practices

Moving between servers best practices
I need to move some websites between two servers which are pointing to the same nameservers.
This is what I am thinking of doing:

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Duplicate site to the new WHM account on new server and then delete the old one.
They are e-commerce sites and don't want to get orders on the old site whilst I am moving it.

Are there anythings I need to be aware of and if there are best practices to follow?
You need to put the billing/order system into maintenance mode for a few minutes, live migrate the website data and then point the nameservers to the new one with a lower TTL setting and then on the new hosting remove the maintenance mode. This should allow all the orders to flow into the new system and if people are still viewing the old website the ordering system will show as offline.
if you have root on bother servers. use Express Transfer from WHM transfer tool. this will change .htaccess of old website to show a nice page informing visitors that the website was moved to new server.
it also change the A records and point them to new server IP automatically .
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1)
I'm using Cloudflare (the free package). Apart from the little extra protection and a sort of CDN (not "real" one for most data, but up to a point, images etc, - paid packages and/or MaxCDN are more "serious" solutions for that), it lets me use Cloudflare's DNS.

That means I don't have to rely on provider's nameservers.
It also means that any DNS (IP) changes get propagated withing minutes, not hours. This part is important when moving sites in my opinion.

2)
Copy all the files, databases etc. to the new server, set it all up.

3)
Choose a time of day/week when the traffic (sales) are the lowest.

Set the "old" site to: "maintenance" (stopping/disabling sales).
Make a fresh "inventory" database copy to the new server.
Change all the DNS records to point to the new server IP and all the other relevant fields.

TEST everything (inventory changes, sales, e-mail sending/recieving etc.).

4)
Light a candle for good luck.
Thanks for all the replies, in my case I should mention that there will be no need to change nameservers or IPs it is just moving to another server within the same hosting company. I am not sure how that will work and which website it'll go to if the website exists on two servers which point to the same nameserver.
In that case it's hardware migration. Your website will be migrated to the new server but IPs will remain same, in the mean time old server will go offline.
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