I use pretty much the same materials in coop and run. Several times/year, I move the build up from under the perches. Just push it out the clean out door into the run. Then, I push the litter from the front of the coop to the back under the perches, cap it all off with some fresh bedding...
I use grass clippings sparingly in the coop. It's a matter of using the right amount. I dump them by the bag full in the run, and the chickens spread them out. Rarely do they stay in a pile deep enough to become the "slime/mold" issue that they are noted for. Even if that does happen, my run...