I just saw a message from Zenos on another thread as follows:

As an example, moving a message from its account folder to Local Folders will result in the removal of that message from the server. Whilst we have not explicitly deleted the message, the act of moving (aka copy then delete?) clearly signals an implicit delete for the server to carry out.

Is this what is happening in my case? If so, how can I stop it from happening and still use local folders?

He means folders under the Thunderbird Local Folder account. Those are folders directly on your computer. Placing message there is moving them off the server.

This does not mean moving to subfolders that you created under the email account that are on the server.

How to stop? Copy instead of move.

Thanks Airmail!

Airmail's approach does solve the problem, but creates two copies of the email - one remains in the Inbox and one in the folder. Is there another solution? I see reference to folders on the server but don't know enough about Thunderbird to follow...

Not is my world. I can drag and drop or use the Move To command and it relocates it to the local folder of my choice and removes it from the gmail server. Suggestion, use move and not copy.

Thanks, but that just takes me back to the original problem

There has to be some way to steer incoming messages into folders without deleting them from the server.

How about you start you own thread with detail of what you are trying to accomplish. The OP was satisfied with the answer that solved the problem and closed this one as solved.