I just hose my run down when it gets really gross, the poo breaks up and dissolves into the soil. If you want, you can deep litter the run. Use straw, hay, grass clippings, pine needles or anything else compostable. Deep litter doesn't work too well in coops that aren't sitting on bare dirt. You need the microorganisms in the ground to really get that deep litter composting. Otherwise, it's just deep bedding. Or you can cover the dirt with sand or gravel, for easier hosing down. I'm less than a mile from the beach, so the soil here is practically like straight sand, no need for me to buy more to cover the run.
I also hose the poop so it soaks into the ground and feeds the worms instead of feeding flies. It eliminates smell and acts as fertilizer. Composting needs water to break down the organic matter but the wet shavings or straw is far more smelly as it decomposes. Straw also grows mold and that can be highly toxic. Breaking down the poop with water eliminates it by allowing it to soak into the soil.
We have a sandy and rocky soil with a very nice topsoil. Keeping the ground clear of bedding material has allowed the rain and regular hosing to break down the poop but it also helps to have a large enough area for the grass to grow because then the grass also benefits from the nitrates and the birds keep the grass mowed.
I used shavings in an area under the nest boxes (we collect eggs from outside the coop) to keep it from getting muddy but the wet shavings caused more problems than the mud so I will never use shavings outside of my planters, where I want mulch to hold the water. When I hose our mud, it breaks down the poop but leaves the topsoil over the sandy soil and it dries quickly.
I have worked on getting rain run off to spread and soak into the ground instead of forming puddles and that has helped keep areas that were once muddy from staying muddy. The worst mud is from decomposing leaves preventing the rain water from soaking into the ground when we get lots of rain so we have started raking up leaves instead of leaving them on the ground. We used to think the leaves would make the yard less muddy but what I have learned is clear ground is less muddy.
The grassy areas are the easiest to maintain so now I am working on getting the grass I destroyed with my landscaping work to fill back in. Having the birds eat the grass actually encourages it to grow faster and root more but we do section off areas where we need more grass to grow in order to allow it to seed and take root before letting the birds eat it down. I have allowed mulch to pile up at the base of our trees and bushes to hold in moisture after the drought we had last summer but the grass will eventually cover those areas too.
As long as the mud is kept clean of poop, it is not going to be smelly so I have found using water to thin down the mud and liquify the waste allows the liquid waste to flow through the soil and keep the topsoil clean. Raking does not allow the layering to occur and it would tear up the vegetation, which is the most beneficial means of eliminating mud.
When birds are confined to a small area, hosing may not work as well. We use shavings in our coops and they get replaced every two weeks with the dirty bedding going out as yard waste. I would prefer to compost them in order to use the mulch but it takes a long time to break the waste down in compost bins so we just compost our grass clippings (we have to mow our front yard) now.
We used to compost our dog poop (we have 4 dogs) with the grass clippings but that did not work out so now we use green cone composters that allow the poop to decompose underground and fertilize our apple trees. We hose out our dog pen after removing the solid waste so the broken down poop fertilizes a bush next to the pen and it also fertilizes our blueberry bushes on the other side of the pen. Dog poop takes longer to break down than bird poop but it also feeds the worms and vegetation. The water used to break down the poop is also good for the vegetation so it is a method that takes less time, energy, and resources to keep a healty ecosystem.
It may take some time to build up a healthy worm population like we have to eat up all of our poop but worms reproduce like crazy in favorable conditions. We also have 3 worm bins and a raised strawberry planter to grow worms in and we use our rabbit poop as worm food and fertilizer as well.