Almost everything related to moving/converting mail accounts can be done using the dsync tool. It can do either one-way synchronization or two-way synchronization of mailboxes. See:
doveadm help sync
doveadm help backup
for more information. Also http://wiki.dovecot.org/Migration/Dsync describes how to migrate mails from another IMAP/POP3 servers.
Mails can be also imported to an existing mailbox using:
The new mails will be appended to their respective folders, creating the folders if necessary. It’s also possible to give a prefix for the new folders, such as “backup-restored-20140824/”.
Mails can be also continuously replicated between two Dovecot servers using the replicator service. See http://wiki.dovecot.org/Replication for more information.