Substitution of Plesk administrated Webserver?

@stephdl @mrmarkuz
Follow up:
The WordPress-Installation with 3 different databases (WordPress, webtrees, Matomo) works fine for testing purposes or future staging. (wp.dargels.de) I also got the mail server on dargels.de running.
An important experience was that Duplicator does not work because it cannot handle 1.8 GB for my WordPress installation. I took UpdraftPlus instead. With this, I could easily restore an already existing backup from the remote storage.

Right now I would do the real migration of myancestry.de
The starting point is that the A-record still refers to the currently productive server. It seems that I first have to switch it off and change the DNS-Rocord to the new server and then continue. This does not allow an uninterrupted migration.

There are a lot of mail accounts assigned to this domain. How do I actually get the existing mails transferred?
Best regards, Marko