I am actually having that problem with the deep litter in my big, open air coop -- it's remarkably dry in there even after a wet winter so my composting-in-place isn't actually composting.
My chicken run compost system is open to the weather, and with normal rainfall, everything stays moist enough for composting where I live. The top few inches are dry but dig down a few inches and I have a composting system that is in that goldilocks' zone of a wrung-out sponge. My chicken run litter is typically 12-18 inches deep, depending on the season, so there is a lot of composting going on under the top 2 inches of dry covering.
We had a terrible drought summer a few years ago. There was basically no rainfall from April thought September. It was really bad. Most of my gardens dried up and died. That was the only time I have ever had to add water to my chicken run composting system. About once a week, I would set the lawn sprinkler to water down the chicken run for maybe 30 minutes at a time. That was about enough to keep everything moist.
It's not just composting that I was concerned about. I was also concerned that if the litter got too dry, then all the beneficial bugs and worms would either die or move on to a better environment. Having 12-18 inches of litter in the chicken run was a big benefit to me because it stored a lot of water under the dry top few inches of litter. I think that if I only had a few inches of litter in the run, it would have dried out every day, which would slow down or stop the composting process, and would not have been a good environment for all the bugs and worms that live in my chicken run compost.
My chicken coop and run is in my backyard, so I was able to string out a garden hose and sprinkler from the house to wet down the run that drought summer. I monitored the condition of the litter and made sure I kept it at that wrung-out sponge zone. No problems for me and the chickens ate fresh bugs and worms all summer while everything else in the yard was burning up and dying.
That drought year was the only time I have ever had to add water to my system to keep it moist and composting. I suppose if you live in dry climate, or if you have a covered run, that would have to be factored into your consideration.