To reiterate, to most sync services, a “rename” operation doesn’t actually exist.
When you rename something on device 1 and then sync, your other devices think you’ve deleted the file, and meanwhile, you created another file with the new name. So, you would see device 2 deleting files, but if you waited for the sync to complete, you’d also see device 2 download the newly-named files.
However, if you stopped this process before it’s finished, bad things might happen. Follow @WhiteNoise’s advice above to avoid trouble.