I don't know of any standard format for exchanging data between email clients.

A first step might be to identify what or how your new email client can import messages, then we can see if Thunderbird can generate a compatible format.

A sure way to do this is to set up an IMAP-connected email account on both systems. Drag/copy your messages into it on the old computer and they'll appear in that same account on the new computer. However this process is going to be slow if you have more than a few MB of email data to transfer.

http://kb.mozillazine.org/Move_to_a_new_PC