Migrate IMAP account between providers - client access only
Try using imapsync, available also as a standalone imapsync.exe. Free, open but not gratis from the homepage.
An excellent tool for Windows is MailStore Home (free)
You install it on your PC
then you Archive all your emails from IMAP Account1 on Server1
you simply Export all your previously archived emails to your IMAP Account2 on Server2
It works also with IMAPS (SSL), and it archives/exports all IMAP folders (or only the ones you wish).
Note: so far I've tried it only to export all messages from one mailbox account to another one. The two mailboxes were on different servers, but both servers run cPanel/WHM accounts, thus both servers' environments were pretty similar.