Kids who can't tolerate milk might well be reacting to the protein in the milk with diarrhea at the symptom. My kids LOVE to drink cows' milk, but they are now so intolerant of it that they get horrible tummy-aches and diarrhea when they do have it.
Also, do be aware that kids who have a hard time with dairy don't necessarily do better on soy; soy is actually a common protein to be allergic to/intolerant of.
My younger daughter loves rice milk (you can get it unsweetened and fortified with calcium and vitamin D like regular milk), while my older won't touch rice or soy milk but will take a calcium supplement instead.
But DEFINITELY do consider taking away the cows' milk for a few days in order to rule out intolerance; with a true intolerance to dairy, the protein can actually damage the gut lining, but if you can rule that out you can breathe a sigh of relief. With diarrhea up to 10 times a day, I would definitely be looking into a food intolerance, especially after 3 weeks of it.