I don't do it. Lots of people do. Seems to be pretty safe though I still don't want to tempt the ammo gods.
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How many reports of blown up tumblers/vibrators have you read about on the firearm forums? If it happened, it would be all over the forums with photos documenting everything.
I've tumbled tens of thousands of live rounds to clean up surplus, remove case lube from reloads, etc, etc, etc.
Some gents over on SnipersHide did a 'mythbusters' style test where they tumbled and chrono'd ammo from 1 hour up to 24 hours. End result, no powder breakdown, no recordable speed differences. Do it, don't worry about the internet lore and hand wringing.