Recommend against Brasso as it contains ammonia and can weaken the brass. I found that wax in dry media works well in protecting the brass from tarnishing.
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No to brasso in the tumbler. It makes the brass shiny but attacks the alloy.
The problem is not that ammonia in itself is bad, but rather that you tend to leave it on after tumbling instead of washing it off.
I like to use a phosphoric or oxalic acid-based cleaner to de-tarnish brass, but it is a wet bath followed by a freshwater rinse to remove the acid.
Ammonia is alkaline.
Either an alkali or an acid will destabilize the brass molecules.
What to do?
Ren has a good idea. Not sure what wax he is talking about, but it makes sense.
When I (dry) tumble, I add charcoal lighter fluid, which is a petroleum product. Thus leaves a very slight, almost imperceptible, oily finish.
Then I store the brass in airtight containers. I suppose you could add a desiccant pack to reduce the moisture.