I am not sure how Thunderbird fits into this. Redirecting your domain is a matter of changing DNS and MX records and that's a job for your registrar or domain host to deal with. If your old and new hosts don't offer any way to transfer your stored email from one to the other, then this is where Thunderbird might be useful. Download everything you want to keep before losing contact with the old server. The Local Folders account in Thunderbird is useful for this. Then if appropriate, move your downloaded messages to your new account afterwards. Is the old account POP based or IMAP based? What about the new account? If the old one used POP then you have a local copy already. If it used IMAP then you do need to back it all up. Another thing to consider; do you have an address book stored with your old domain provider? Does this need to be rescued/preserved too?