Can you tell me how you got started on this type of run bedding? My run is half sand (covered side) and pine pellets (uncovered side) which just turn to hard dirt/mud and need to be raked regularly. I love the idea behind the composting run, but am having trouble picturing how the conversion would go- will the system still work if I just start putting grass clippings in during the spring? I was also curious if it retains a lot of water. My run is half roofed and half open to the elements.
First of all, my run is completely uncovered (it has bird netting for protection, but no roof). If you want to make compost, you need to have the litter exposed to rain/water to help the process. If you have a covered portion of the run that you want to make compost in, you could probably spray it down with a garden hose every so often. The ideal is to have the compost litter about the moisture level of a wrung-out sponge. The top inch or so of the litter might be bone dry, but underneath it should have that wrung-out sponge consistency.
I started out my chicken run compost system with a base layer of wood chips over the dirt ground once the chickens ate all the grass. Then, in the spring, I started dumping my grass clippings in the run on top of the wood chips. You want to be sure not to dump the grass clippings in a big pile, because they will heat up and smell really bad. I spread out my grass clippings and/or put them in small enough piles that the chickens will spread them out as they scratch and peck through the grass clippings.
My chickens like to eat some fresh grass clippings. Because I have hawks and Bald Eagles overhead all the time, I cannot let my chickens free range for grass and other greens. So, I bring the free range to them in terms of fresh grass clippings.
When I pull weeds from the lawn or garden, they also get tossed into the chicken run. The chickens might eat some of the weeds, and the rest gets mixed into the litter to become compost.
In the fall, I mow up my leaves with the riding mower and dump the leaves in big piles in the chicken run. It does not take long for my chickens to spread out the leaves and make everything level on their own. Evidently, there are lots of good bugs in all those leaves because my chickens will be scratching and pecking like crazy in those piles of leaves.
As the chickens scratch and peck through everything in the run, they continually mix the litter which aids in breaking down the compost even faster. By the end of that first summer, I was able to harvest wheelbarrow loads of compost that I put on my gardens to winter over. Since then, I have so much compost ready to harvest that I have given some away to neighbors for their gardens in addition to having more compost than I can use myself.

I had so much compost to use that I built another 6 raised garden food beds to grow people food for the family. This spring, I plan on building another 4 new raised beds, filling it with hügelkultur wood on the base, and then putting about 8 inches in the top with topsoil and chicken run compost mixed 1:1. Makes great raised beds.
Of course, I now have enough compost to top off my existing raised beds with chicken run compost in both the spring and fall.
If I had to buy the compost in bags that I now harvest from my chicken run, it would cost me hundreds of dollars every spring and summer. That's why I tell people I have composting chickens and get eggs as a bonus. So, I now grow more people food than I ever did before thanks to my chicken run compost but no longer go broke buying big box store compost in bags.
My chicken run never smells. After a nice rain, it smells like a forest floor, which I like. If you ever have any bad odors in the chicken run, you need to add more carbons like wood chips or leaves. I used to worry about the grass clippings clumping up and smelling as they decay, but if your chickens spread everything around like mine do, then you will have no problem. The uneaten grass clippings will dry out and mix in with the litter, making great compost.