I tested both in /home/user and in /home/user/MyDocs. I only published the former as the latter matched chill's experience.

I suspect some kind of caching issue. I do not believe it is busybox's fault, I assume it has more to do with the kernel and its FAT/FAT32 support. My first guess would be that the FS somehow caches the names. It might be interesting to figure out if there is a timeout or another way to reset it. I tried very briefly cd to another folder and back before the last mv but it did not make a difference. I also tried closing and reopening the terminal and that did not make a difference either, confirming my suspicion that busybox is probably innocent.