I have had the same problem with tires on my york type rake (Arps). The problem was the bead, not the tire carcass. I deflated the tires, and cleaned up the bead with sandpaper and painted them. Then I put rubber cement on the tire bead where the dirt and gravel had abraided it. Do this all without dismounting the tire, one side at a time. Ultimately, the problem was that the wheels did not swivel very well so they dug into the driveway dirt. When I fixed that problem, the tires stayed aired up. There might be a method to fill them with foam, but I would start with the basics. If the tires are punctured by nails, thorns, fence wire, etc, then the slime may save you. Check Shraeder valves, too. and good metal stone caps that seal.
Lastly, since these tires are usually small, fill them with salt water by sinking them in a bucket. Add air to pressurize them. At least then you will know where and when the leak occurs. The salt water is good for dust control on your driveway !