On your old computer, Quit Thunderbird then open your Thunderbird profile folder. Inside the Mail folder, you will find another folder named after your mail server, e.g. pop.example.domain. Copy this folder to an external drive and rename the copy to OLD. Connect the external drive to your new computer.

On your new computer, create a new folder in Thunderbird, under the Local Folders account. Name it OLD. Under the OLD folder, create a sub-folder and name it Temp, then Quit Thunderbird. Open your Thunderbird profile folder and go to Mail > Local Folders. Inside you will find a folder named OLD.sbd, open it. In your external drive, select everything inside the OLD folder and copy them to the OLD.sbd folder. Launch Thunderbird and all your old messages will be available in the OLD folder under the Local Folders account.

You can leave them there and access them from there when needed, or you can move them to your inbox to have them all in one place. If you choose to move them, select no more than 2000 messages for moving at a time. You do not want Thunderbird to choke on moving too many messages at once.

I have a similar situation, except both the old profile and the new profile are on the same computer. (Background: My ISP has required that all my domains use the same POP and SMTP servers, rather than different ones for each domain like I used to do. I messed up my configuration trying to do that, and had to start all over. One of the reasons I am re-organizing is that my directory structure and the files in it have been passed down from generation to generation all the way back to Eudora 2 days... Lots of cruft .... hard to tell what's valid.)

• So, rather than copying to a portable device, renaming, then copying back to a location under LocalFolders, shouldn't I be able to simply move entire directories from the old profile to the right place in the new profile (keeping in mind the server names have changed and some things need to be renamed)? I have tried to do that in the naive way, but when starting the "destination" TB, I only see the directory structure, not the messages.
• How does TB decide which subdirectories it should inspect for valid emails , and which not? This might tell me where I have to rename things.
• How are .sbd files generated; or what is their structure?
• When I am done restoring this computer's TB, I would like to merge the TB files from some of our other computers. The most notable differences among them will be in the "sent" folders, but also, some of the configurations (e.g. mine) have more sophisticated filtering and message organization (than, e.g. my wife's), which I could like to install on the other machines. Knowing how to quickly merge folders from different profiles would be very helpful. as removing duplicates is easy).
• It would also be nice to propagate password changes among the systems...

(Yeah, I know, Maybe I need a local IMAP server.)