I've used a vibratory tumbler on brass for years. I have experienced the staining you describe on various kinds of brass for years. It's nothing to worry about, in my opinion. A lot of time, rifle brass doesn't need tumbling, but rather just a wipe down with a paper towel. Sometimes I will use alcohol on a paper towel to clean with.
Reasons to minimize tumbler use are 1) high concentrations of airborne particulates, particularly lead, and 2) build up of "stuff" on the inside of the case, leading to (potentially) high pressures.
So, clean and tumble as required, but don't get too concerned with having the brass bright enough to see yourself, in my opinion.
Hope this helps, Tom