Best approach is to setup all the email accounts on the new host keeping the same passwords for all accounts if possible.
Once you change the
DNS to point at the new server ... things will just changeg over ....
However depending if the client is keeping all the old emails on the server ... you may need to copy them first ?
Assuming that the mail server is running from mail.domainname.com and both hosts have the same setup ... things should just transition smoothly