moveimap is a tool for migrating IMAP email data from one server to
another.  I wrote it when I did a massive migration of ten years of
email (about 2 gigs) from my home server into Google Mail.

It's designed to be robust in the face of failures, as you are likely
to encounter a lot of failures during the multi-day process needed to
migrate 2GB+ of email into Google Mail.

There are probably much more user friendly tools out there now, but if
not, give moveimap a whirl.