There is an add-on to TB which allows import and export of emails in various formats to a separate storage medium (external disk, thumb drive, etc.).

The instructions on how to download and install it can be found at http://barryduggan.info/exportMail.php

Hi, I have gone on the site, clicked onto add-ons. All that is there is a back-up mail...is that what I require? thanks

I am proposing that you back up your email using ImportExportTools in a form which can be imported later. Then after you have switched to the new domain/host you can import the mail.

If the domain/host is only an ISP and not your email provider, I don't think you will loose what you have on Hotmail, but it's safest not only to back up your mail, but your profile as well. Look at https://support.mozilla.org/en-US/kb/profiles-tb#w_how-to-find-your-profile