Are there any benefits/negatives to putting a few shovelfulls of compost (no chemicals - I'm all organic) in the run?
Compost retains moisture so it would tend to make the run wetter/muddier. I also have uses for all the compost I can get my hands on. I really can't think of any benefits other than the bugs that would be in it and I consider that insignificant.
I toss my kitchen waste in the run for the chickens to eat. Once a week I rake the whole run and make a pile in the corner. My run is sandy and I add DE often. In the spring I use everything that has accumulated in the pile for the garden.