The IMAP Sync utility inside the Plan Manager handles both migrations into SMTP Zen and exports to another host. You do not need extra software - just the login for each mailbox involved.
You can close the modal after it is queued - the transfer keeps running in the background. Larger inboxes take longer; check back later to confirm status.
Exports are handy when you are consolidating accounts or giving a third party a full backup.
Using IMAP Sync keeps history intact so users see the same conversations no matter which provider they use.