If the messages didn't transfer, it may have been that you were using thunderbird at the time of the copy. If the old Mac is still available, I suggest looking in the Mail folder (assuming a POP account) of the profile to locate the message files. You can do that on old Mac by clicking help>troubleshootinginformation in thunderbird, scroll down to 'profile folder' and click 'open folder' and then clicking on the Mail folder and then clicking the account folder. If your missing folders are there, Exit thunderbird and copy to external media. Use the same steps on new Mac, and copy the message file in (BUT, if you have been retrieving new messages in the interim, you should copy to the Local Folders folder instead to avoid overwriting the new messages.

Saw this yesterday, but lost it, so here I am again. I cannot open Thunderbird on the old Mac.

I cannot advise on the file layout on a Mac, but if you can locate the profile, the above suggestion still applies.

Thanks for your help, David. I had trouble getting back to this page, hence the duplicate. Can't find the profile and cannot open Thunderbird at all. I'm going to find someone I trust to use TeamViewer and see if he can unravel the problem. Thanks all the same.

If the Mac has the ability to search for specific file names, search for abook.sqlite. That's the addressbook and if you can locate that, you've found the profile. Good luck.