Are you concerned about the inside of the die? Or the outside? The inside cleans up quite nicely with a dowel and a piece of cotton wetted with Hoppes No. 9. The outside cleans up quite nicely with a good wipe with a cloth dampened with isopropyl alcohol. I wouldn't think that the tumbler would hurt the die (especially if you removed the decapping rod).
Spray them with a product called Rust Prevent available from most online reloading suppliers. I put that on mine when I lived on the coast and it stopped that problem.Rick.
I do regular maintenance on all my dies, especially those that are hardly use. I swab the inside using bore mop soak with Break Free or WD-40. I then spray Rem-oil on the die body and wipe it with clean with cotton cloth before storing.