For wet tumbling with stainless pins I have had success with a tablespoon of washing up liquid. Added after the water so you don't create a big cloud of foam. Tumble for a few hours then rinse with plain water once or twice, then separate and air dry. Tried adding a little citric acid to the mix, but it didn't seem to make any difference for me - but my water supply is from a well and not perfectly pH neutral anyway.
Personally I do it (when I bother to clean brass at all, which is rare) after 100% of the other brass prep so that any remaining lube is removed along with everything else. I'm not saying that's "right" or even the "best" way to do it, but there is something nice about using brass in that condition - all other things being equal your reloads will look better than a lot of factory. If there's one genuine practical benefit it's that it will get the primer pockets clean without the risk of enlarging them (assuming you at least deprime first anyway).
The tumbler I use is the Frankford Arsenal one. It's pretty decent to be fair and has been very reliable - although as I say, I almost never use it.