Your idea is to allow customers to enter the credentials of their existing mailbox and have imapsync copy over the messages?
Yes it would be ideal if the target could be pre-filled from enhance and the source entered by the customer, have it copy over the folders (structure is as important to some as contents) and the messages within them. It can take some time though so it'd need to spawn itself as a task with very long timeout.
Not so bothered about the other mentioned tools, they might be nice, but moving that stuff manually is no big hassle
True, I was thinking perhaps having 'Website > Advanced > Import' listing email/database/ftp transfer options so they're more exposed and clear to purpose, maybe the database part being SSO to $PMA/index.php?route=/server/import would be good enough there.
I quite liked the idea of a PHP script people upload to the losing host and entering a user enhance API key to automate the migrations but I figure they might run into memory/resource/timeout limits on the remote side.