Yes, it's common for people to add a little mineral spirits to media for polishing. But if you have enough dust to build up like that, though, I don't know that you could add enough to prevent it. If you get the media too heavy with liquid, a vibratory tumbler may stop vibrating well.
What folks usually use for cleaning lube off cases is plain (no polishing compound) corncob media. That's because it's much more absorbent than walnut. You can get 40 lbs from here
for less than $1 a pound. That's more than you'll pay if you have some place that sells it locally, but because it includes shipping, it is cheaper than most other outside places. 20-40 grit is the size Lyman sells.
I'd at least clean the walnut media, as FrankenMauser suggested, or look at just replacing it. That's cheap to do with pet store walnut litter, or if you buy the corncob, use that, separating the corncob you've added polish to from the corncob you use to clean lube off the cases. Mix some mineral spirits into a tablespoon or two of automotive polishing compound to make thick liquid to serve as your polish, and add to it the media, let it vibrate into the media with the cover on for 10 or 15 minutes. Voila! New media.