Thanks for the great reference!
I made this a bit more complicated by not importing all old clients/sites from the old server and by changing one client on the new server. This obviously caused more trouble than it was worth, most notably with mail. So much for trying to save time.
I finally resolved it by just manually copying the actual mail files from each folder within /var/vmail into the new respective folders.
I would recommend to anyone doing a server migration to just copy the users/mail/web as is and then remove whatever you don't want or make client changes AFTER the migration instead of trying to be a rock star and do it all at once, it would have made life much easier for me.