I usually tumble brass before resizing to reduce possible scratching of my resizing dies with dirty brass. I wipe the brass with a towel to remove the abrasive compound dust from the outside of the case. It just occured to me that the dust left on the inside of the case-(I do not wash cases after tumbling)-is being blown out of the barrel. I use Lyman green corn cob media and brush the inside of the neck with a brass brush before resizing.
Is there enough residue from the media to cause any damage to the barrel?

Not likely. Most of the polishing media is much softer than steel. In many cases it's diatamaceous earth which is quite soft.

Any polishing residue left in your brass will certainly be traveling down your barrel at 2,300 + fps under intense heat and pressure when you squeeze the trigger. Diatomaceous earth and other polishing compounds are relatively soft when compared to steel, however, if you can avoid shooting this stuff down the barrel, the better off you are. We take great pains to avoid introducing any abrasive into our barrels, it only makes sense to avoid shooting polishing residue down our bores as well.

After you tumble your brass, it is a simple matter to rinse your brass with hot water and run them into the oven for a few moments to dry them. It takes an extra 15 minutes to do this, what do you have to lose?

Zero Drift; Thanks for the reply, I'll rinse the cases.