Server migration

[Gandalf Corvotempesta]

Hi to all.
It's time to migrate an old server to a newer platform

Some questions:
1) what happens by changing the pop3/IMAP server on the client?
Is the client (Outlook, Thunderbird,...) smart enough to not download every
message again?
I'm asking this because the easier way to migrate would be move all
mailboxes to the new server and then change the hostname on the client

2) what if I add a dovecot proxy on the new server, proxing back all
requests to the older one, if the mailbox is still not migrated?
Would the whole pop3/IMAP transaction happen through the proxy or there is
something an http redirect (or anything similiar to the SIP protocol) ?

3) I think the response to this is no: is dovecot able to log the hostname
used for the connection? I have multiple domains pointing to the same IP.
Something like the Host header in Http.