mail is stored in mbox files. These along with an MSF file make up a Thunderbird "folder". Sub folders are stored in a directory with the name of the parent "folder" with .SDB ending the name.

So the sub folders of inbox will be stored in a directory named inbox.sdb

For your purposes you need only the mbox file which actually has no file extension at all and to be aware the sub folders will be in the SDB directory. Copy them into the new profile in the "Local Folders" directory in the Mail directory and when you start Thunderbird they will just appear in local folders. Note this copying does not work with IMAP mail accounts.

Looking in Mail > Local Folders > Inbox.sbd, I do not see a file named "mbox". I see a dozen or so folders named "XXX.sbd" and a bunch or files that have directory names with no extension and a duplicate file "name.msf".

My first try after thee .zip failed, I copied the entire Local Folders directory to our server then downloaded it to the new machine.

Looking in Mail > Local Folders > Inbox.sbd, I do not see a file named "mbox". I see a dozen or so folders named "XXX.sbd" and a bunch or files that have directory names with no extension and a duplicate file "name.msf". My first try after thee .zip failed, I copied the entire Local Folders directory to our server then downloaded it to the new machine.

You did not read it like I though I wrote it. The mbox files have no extension.

Personally I just copy the entire profile folder complete from the old to new machine. Calendars contacts passwords and probably a load of old rubbish as well are included. But I also get everything I had before. The profile import looses passwords, and it is a big deal for me to go off looking for 20 or 30 passwords, just because the profile importer was released broken.