Ultrasonic, dry, inspect for any missed gunk, shoot with Hornady dry lube, back in the press. Dental picks, bore brushes, gun brushes, compressed air, solvent, you name it available if needed, but it never is.
Biggest thing I do to enhance my die cleaning is to never work with dirty brass. I decap in a universal decapper then clean. My dies only touch clean brass, so all I have to deal with is built up case lube, which isn't a thing for my carbide pistol dies, and is pretty easily avoided in rifle dies by attentively and selectively lubing. Using the same dry lube for case lube as I do for die maintenance is an advantage too.