I know (waving hand in air). Sawduat will rot and turn to gunk and driveway will have to be scraped and more stone added. Only thing to add to such a driveway is more stone. Agree with above - horses need to have hard surface for a multitude of reasons but hoof health is one of my top reasons.
Maybe adding STONE dust to an area or path, that you use consistantly, would be the best idea. Large, sharp crushed stone, cobbles, can be slippery or painful to some horses, lamanitic horses, thin soled horses. You don't like hurting them when they must cross the driveway.
Getting a much smaller diameter of stone, dust or chips, and making that path by laying the dust or chips over the larger rocks, would be the best choice. Driveway will still be hard for vehicles, drain well, with no clogging like sawdust could cause. I am presuming a limestone type, crushed rock is your present driveway, and they can be real sharp rocks.
Using organic material like sawdust, means it would be constantly breaking down, turning into dirt. Organic material does that. So to keep rocks covered, you will be constantly adding more sawdust. The dirt produced will clog up the large stone's ability to work as drainage, promote weed growth in the drive. Cheap, fast and dirty solution. Works in emergency, but will need cleaning up and a redo of driveway area over the long run.
With stone dust, you can add it over your present stone driveway, cover the big rocks with tiny rocks that sift down around and fill the spaces. You can tamp the dust down, using a power tamper, which does a good job of packing things tightly. Should get the stone dust firmly in place to make a firm surface to use for the horses. I would water after tamping stone dust, see if things move, fill again with stone dust, tamp again with more dust on if needed.
You want the stone dust to raise the surface so horse won't feel the bigger rocks underneath. Make path generous, at least 4ft wide, with dust a couple inches thick when done tamping.
We always rent the power tamper for any stone work, does a much BETTER job than any hand tamping you can do. Takes much less time to do that better job. Moving the stone itself is tiring without a bucket on a machine, so save your body if possible.
Have the stone dust delivered, dumped onto a tarp, right BESIDE the driveway where you want the path across. Less work to move the stone for you.
Talk to the place you will get the stone, tell them what you want to do with the stone dust, how it needs to fill in over the other rocks. Local names vary on rock choices. What I call stone dust could be "fines" or a sized number stones are crushed down to. Could be from real DUST to tiny rocks the size of cupcake sprinkles or penny size rocks! You may need to visit the place to choose your stone dust, see what you are actually talking about with them. Better to order too much, have left over stone dust than to come up short. Any extra can come in handy to fill the mud hole in front of the barn door, a wet spot or puddle that is always in the way. You only have to pay one delivery fee then.