Moving mailboxes from Courier/Maildir to DirectAdmin/dovecot/Maildir

I have an old mail server running Postfix and courier-imap. We want to split our customers off onto their old server, preferably something with a pretty pointy-clicky interface so that they can manage their own accounts. (Yes, people do still buy email service these days.)

The old server runs FreeBSD, postfix, and courier-imap. The new server runs FreeBSD 9.2 with DirectAdmin. DirectAdmin is a canned package that puts a customer-friendly front end on a whole bunch of standard Unix software. (W...