I haven't tried a combination, but a while back I took to washing my walnut media periodically, which I think basically achieves what you're after here.
I've had this conversation a few times with people since and, in hindsight, find it funny people are critical of walnut (or whichever) media for not cleaning well enough when they are/were re-using the same batch endlessly.
If you wash a window with a dirty rag, no one is surprised that the window ends up looking dirty. Yet people expect tumbler media to just magically send grime off into the ether
If you're not cleaning your media as well, at some point you're obviously just re-circulating the lube and grime from the cases.
Check out this which I posted a while back.
Before and after washing tumbler mediaThe results of the cases before and after the wash were night and day.
Hope that helps.