Assuming the server you are migrating to has Plesk installed, and assuming you have root access to the old Plesk server, you can use the Plesk Migration Manager. If you've used the cPanel Transfer Accounts function it's basically the same.
To avoid downtime, transfer the site, test, plan on a time to change DNS off hours, then re-transfer the databases and latest emails manually. If you don't know how to do the last two steps, just delete the domain from Plesk and re-transfer again just before changing DNS.