When moving files from a folder to another, the files are moved correctly (and no other operation is needed) only if the device is online.
For example, I moved about 20 files from a device while all my other devices were offline (and to be clear, those devices were up to date prior to the moving). The operation was performed on the Dropbox website (because I was hoping Dropbox would be smarter, but alas...) and the online device performed it correctly. However, when I started my other devices, they all had to resync (which is expected) but then for whatever reason, they needed to download 4k files (basically it downloaded the entire folder again).
So my suggestion would be to use a system close to what VCSs do: keep track of the operations that are done on the online devices, so that offline devices can replicate the operations instead of downloading entire folders.