I always tumble with the old primer in place. Then when depriming/sizing any media stuck in the flash hole gets knocked out in one go.
Used to tumble after depriming but media (corn or walnut, doesn't matter which) gets stuck in the hole and depending on the media can get really jammed into the primer pocket itself. Had to go back a second time and either knock it out with a punch or run it through a decapping die. Seemed like unnecessary effort when it could all be taken care of in the resizing operation.
Second advantage as already noted is that a clean case is run through the die.
Never had a problem with media damaging a decapping pin on any die - RCBS, Redding or Lee.
People forget that the main reason to tumble cases is to have them clean so grit does not damage your sizing die.
I used to tumble, deprime, and tumble again to remove lube but got real tired of picking media out of cases and flash holes. I've switched to water soluable lube and simply rinse the cases after sizing.
