I find that tumbling first then depriming is best for me.
I don't like running my cases through my dies until they are clean. I scratched up a sizing die rally bad and would rather not repeat that.
I don't like picking debris out of flash holes either. To me it's a real pain.
I did buy the RCBS case prep station and it's very easy to clean up pockets with it. It's not cheap and not for everyone but I like it.
I just started blackpowder cartridge loading and so far, depriming with a Lee hand press and a universal deprimmer die and dropping them into a jug of soapy water is working fine. I rinse them when I get home, or back to the RV, then let them dry. Maybe a little wiping off with a rag and I'm ready to reload. I don't have a need for bright shiny blackpowder cases if it means tumbling with empty primer pockets.
Like I said, I'm just starting BP cartridges so I'm sure I have room to learn.