Transfer the site to another server

Contact the professionals, they are quick for the coin will do =)
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In fact the issue: 1) move the site 2) add an a record in DNS

Resolve via DNS? Not?
domain.com IN A xxx.xxx.xxx.1
sub.domain.com IN A xxx.xxx.xxx.2

About turning to the pros — and invaluable experience? )
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In fact, the question: will the website transfer, domain Registrar change the DNS address, then "add existing domain" to a new hosting, and to create A record for all subdomains on the old server?
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Did I miss anything?

You would have said what panel is used on the website, then it was possible more effective advice to give is "move files to change ip"

Copy the site database, to delegate the domain to the normal NS (with muzzle ispmanager, or regressive or their partners), correct Dnsy. Everything in General is.

For all subdomains in DNS to specify the IP of the old server.