user data transfer when migrating to new server

I;m setting up new server, what are the options for transferring user's mail ?

I've tried imapsync, that seems to work well, but, what options do I have for doing accounts or domains where I don't know user's password ?

last time I've done, some users had lengthy period of re downloading 'old' mails from new server, is there a way to overcome this ?

thanks, V