I presume they are all set to use POP?

Yes, all are POP.

On the misbehaving one, it has 15 accounts, the misbehaving typically has around 5,000 emails in the inbox and 40,000 in the trash.

The misbehaving started at the same time that Thunderbird quit grouping the accounts together so that you had to select "trash" and then it showed all the trash accounts, one after the other, "Sent" grouped all the 'Sent' accounts together, etc. This was an excellent way to do it. Now, all the accounts are carried totally separate which is a royal pain in the rear.

Right click the account in the folder pane. Select settings. change the server setting to leave messages on the server.

Right click the toolbar at the top of the screen and select menu bar On the menu bar select View > Folders > unified

Thanks. The last part of your response fixed the folders problem.

As far as leaving the email on the server, that flag has been set for all 15 years or so we have been using Thunderbird. Only recently has it started misbehaving. I have even unset the flag and set it again thinking that might make it start behaving again; all to no avail.

When I am at work, I send an email to my email (@mindspring.com) expecting it will remain n the server until I get home and retrieve it (home deletes email from server, business doesn't).

I do the same thing from home, but to a different email address (@mybfl.com)  which works fine (business deletes email from the server, home doesn't).

In the last couple of weeks, the business to home email (@mindspring.com) does not work because the emails are getting deleted, before I get home, by the business invocation of Thunderbird which typically runs 24 hours a day, 7 days a week. It appears to have started with a particular update of Thunderbird but I have no idea which one.

Did you perhaps add the mail accounts to your phone as well about when the problem started.