How can I get Thunderbird to automatically move messages?

Solution 1:

You can use message filters tool of ThunderBird.

  1. Go to Tools -> Message Filters
  2. Add New Filter. Select a name, a type of match and the perform action (destination folder).
  3. Select the source where the filter will run.
  4. Run now.

You could create the filters that you need.

Greetings!

Solution 2:

Message Filters Button add-on will ease the problem. It is similar to the "Run filters on Subfolder" add-on.

This issue bothers a lot of people: Bugzilla thread regarding this bug.