you COULD use imapsync actually… but it failed me with some obscure error.
rsync is NOT a good idea… because dovecot/thunderbird/icedove does not rebuild index files/folder structure.
the solution is actually simple: use thunderbird/icedove your mail client!
- create same mail account on new server as on old server
- add old and new account inside thunderbird (you can use IP addresses if the domain is already pointing to the new server)
- drag and drop the old mails-folders from your old account to your new account
- wait… all mails get downloaded locally from your old account and then uploaded to your new account, depending on your internet/amount of mails/server-speed this could take a while.
HINT: You may need to do this serveral times (first it just copied/created the folder structure, but not copied the mails)
so i had to manually drag and drop (Ctrl/Strg+A = select All really helps) all mails folder-by-folder from old server-account to new-server-account.
BTW: SERVERFAULT.COM SUCKS! i already know they will remove my answer. so here is the screenshot: