My chins have the same problem. I've read that you could get an unscented wipe and wipe it off a bit, then when the fur dries give her a dust bath. A dust bath alone gets rid of most of the stain on mine.
I've been wondering this same thing!
I recently just got my second chin and she is white beige in colour which is new for me, as my first chinchilla is standard grey.
I was very alarmed today thinking that my chin might possibly be experiencing some diarrhea, but if that were the case she would probably smell and not be so lively, right?
I cleaned their cage and didn't notice any issues though.