You can use cross-platform imapsync utility for this purposes. You have to create empty mailboxes on destination server. Before deleting mailbox on source server check everything twice on destination server (i mean is everything really synced). You can ask user to use new mailbox for some time (e.g. one week is enough) and then if there is no feedback you can do with source mailbox everything you want.
Example command:
imapsync --dry --noexpunge --no-modules_version --host1 source.server.com --port1 993 --authmech1 LOGIN --user1 source.mailbox@source.server.com --password1 password.on.source.server --ssl1 --host2 destination.server.com --port2 993 --authmech2 LOGIN --user2 destination.mailbox@destination.server.com --password2 password.on.destination.server --ssl2 --syncinternaldates --split1 100 --split2 100 --useheader Message-ID
Twisted Linux 