You might want to look at this article about muddy runs. You might get some other ideas that can help. The best ways to handle a muddy run are to either keep the water out to start with (good luck with that if it sets in rainy) or make sure it will drain out once it gets in.
Gravel can help though it is usually a temporary fix. It will normally work its way down in the mud and disappear over time. How quickly it disappears depends in the soil and how muddy it is.
It's best to not use sharp gravel. There is a big difference in what can happen and what will, but it is possible a hen could cut her foot on sharp gravel. If that cut gets infected, you have what is called bumblefoot.
I suggest you look for rounded gravel, usually found in a riverbed. Over time, the water has taken off the sharp edges. I don't know how much you need or what is available, but I've dumped a bag of pea gravel in front of the coop door coming in from the run. It lasts a little while and really dies help. You can probably get a dump truck to deliver a load of river gravel if you need that much.
Another advantage of dumping gravel in there is that they then have a good source of grit. They'll pick out the bits the size of a pea or smaller and eat it for use as grit.
I just finished building my run and decided to put in fairly deep sections of Stone dust or screenings, I used old cedar fence posts to section the areas off and then kept them in place with stakes, this way it won't wash away and the chickens have dry areas.
I tried stone dust and found that it compacts. I changed to river sand and the ground is not so compacted now. Sand generally drains well, is easy for the chickens to dig through, and is somewhat easier to move than gravel.
Thank you for the helpful link, Ridgerunner. I'm in the planning process with my coop and run, and found a number of ideas there that will work in my situation. A good bit of this part of the valley is flood irrigation. Since my property sits lower, and my neighbor who doesn't seem to understand that all the water in the world doesn't help pasture that's being overgrazed floods constantly, I need to devise a way to keep my girls outta the mud. Exactly what I needed.