I added new soil and stirred things around when it was looking in need, but never actually dumped anything til they were in the coop. Then the whole lot went into the compost. I never had much of an issue with smell - though by the time we started using the soil, the brooder box was in a well-ventilated shed. We had quite a bit of chick dust built up on the items stored in the shed, too - but at least it wasn't built up in my laundry room!
I wish I could remember where I saw the idea - I think it was a suggestion in The Encyclopedia of Country Living by Carla Emery (always a great resouce!).