Synchronisation software to find files that have moved paths within a folder?

"It all depends on how you set up the sync process.

Most sync programs have options on what to do if a file is/isn't in the source or is/isn't in the target. Properly set to completely echo a directory, a sync program will delete files in the target that are not in the source, and will copy from the source files that are not in the target or that are newer.

However, no sync program will detect that a file was renamed, in the sense that it will not rename the file in the target directory. It will just delete the file with the old name, and then copy the the file with the new name. This will take more time, of course, but sync programs don't have artificial intelligence.

That said, my own favorite sync product is FreeFileSync (go to end of the page)."

"I heard Microsoft's SyncToy program has this capability.

or this tool by Risefly."

"Syncthing and git-annex should do the job

Git-Annex is as the name suggest based on git, moving and/or renaming a file will be recognize and so not duplicated.
Syncthing will also sync the modification instead of duplicating the file

Both of those software are also able to keep version of your file and/or to keep deleted files for a certain period of time.
The main issue is you use a backup software while you tend to synchronize files, this is why you/we need a good file planning (how to store and how to name files) before starting backup and storage."