Creating a setup based on the sticky thread gets proper subfolder syncing of added files, but it doesn't track deleted files. The rule I created before reading through the forums properly tracks additions and deletions, but doesn't delve into subfolders (which makes sense since the Modification Date of the subfolders doesn't bubble up to the folder the rule is attached to).
This is the rule I built:

What do I need to change/add to get it to run through all subfolders of Dropbox/Backups? When I tried a test setup that made use of Run rules on folder contents that Mr_Noodle explained in a different thread, it would do the right thing for additions (but ignored deletes, unfortunately) except it would flatten all the contents of the subfolders into the top level of the sync destination; not good.