For most of my work I do, I'm using Linux (often Ubuntu); so most of my experience is based on Linux. For this environment I gurantee, that a simple rsync command works great. To be safe, you just close your Thunderbird before you start copying. Then you can rsync like this:
rsync -av $HOME/.thunderbird /PATH-TO-YOUR-EXTERNAL/DISK
where /PATH-TO-YOUR-EXTERNAL/DISK is the path to a previously (auto-)mounted external disk, used to transfer your mails to another Computer.
While $HOME/.thunderbird is the default directory Thunderbird uses to store it's data, your directory my differ.
When you receive new mail, essentially this new mail is appended to your index file (if you use mbox storage format), or it's stored as a new small file (if you use maildir storage format). In either case, rsync copes very well with added/deleted emails and only copies changed files, and only changed blocks of these files. A huge Inbox won't prevent a fast rsync this way.
While rsyncig mailboxes this way has always worked flawless for me (using Unix-ish OSes), part of your question makes me assume you want to copy your mails from a computer running Windows. There are versions of rsync for Windows - there was at least one about 6 years ago, when I used it for backup copies (sort of). It worked, but it had some issues, because the version I used was a blind port from Unix rsync, and it did not find all the file attributes it expected. But it worked, producing some error and warning messages. So I can't guarantee for that case, but it's worth a try.
By the way: when you configure Thunderbird for a new account, I recommend to use maildir storage format. It uses a little bit more disk space (at least on block for each mail, even it contains just a few words). But it excels when you move or delete mails, as it just moves or delete files and has not to manage 10GB file or the like.
Greetings, TomTomTom
