In my chicken house (which doesn't have a dirt floor) I start a "deep litter" by putting in a layer of dirt from the garden, then use wood shavings with that. I rake it at some point during the day and add more shavings as needed - mostly just under the roost. Since it's inside and not a dirt floor, I just take some out from time when needed. This same method can be used no matter how large or tiny the house. I just leave a little of the old litter and start again - sometimes adding dirt and sometimes not.
When I take the litter from inside out of the house, it goes right into the run. It makes a wonderful, deep litter and keeps the outdoor run from becoming hard, impacted disease-breeding ground.
I had a tree company chip some wood for me (they'll leave it free usually for the asking). I originally let that set in a pile for about 6 months to "cure"...it became full or red worms and other little bugs by that time. Then I take that over to the run by the wheelbarrow-full and dump it in a pile. The birds LOVE digging through it and spreading it.
Between the wood CHIPS from the trees and the wood SHAVINGS from inside, it makes such a healthy ground! The birds can dig through for worms at any time. Never becomes unhealthy.
Last year 2 of my daughters came over and dug out pickup truck loads to take to their gardens from my run. Just under the top layer is the most beautiful earth - smells wonderful. That combination of the nitrogen (poo) and carbon makes such great soil. And a wonderfully healthy place for the birds.
My birds aren't confined to a run and are only closed in now and then, but if I was in the city or had limited area and they had to be in a run, this is one thing I'd do as a natural/health-building environment for my birds. I think it's probably one of the best little management methods out there.
PS: I only use wood CHIPS and leaves in outdoor runs; shavings inside.
ETA: This post shows what the chips look like in the run:
https://www.backyardchickens.com/t/...eeping-thread-ots-welcome/37870#post_16535295