I do it all the time.
I have an on-going test right now, with some ammunition that has been tumbled for 28 hours. When I feel that it has reached an adequate point for testing, I'll shoot most of it, and pull the bullets from a few cartridges to inspect the powder. I don't expect anything out of the ordinary (other than sparkling-clean case walls inside the case).
Tumbling with lead bullets, however, will pick up abrasives in the media. It can get a little messy, and you probably don't want to send abrasives down your barrel intentionally.
"Such is the strange way that man works -- first he virtually destroys a species and then does everything in his power to restore it."