I think it may well be related to diet - and asthma, allergies, and food allergies/sensitivities are all related to diet. As our diets have become poorer (through processed foods but also through chemical exposures, poorly nourished beef cattle & chicken, and fruits/veggies grown in depleted soils (and I'm not even counting the genetically modified foods), our percentages of allergies and asthma and digestive issues have risen dramatically. Ask any kindergarten teacher what's happening in her classroom - she has nut-free classrooms, kids with epi-pens and inhalers, diabetes, bi-polar disordered, increasing rates of ADD/ADHD. My friend has been teaching for 25 years - and 25 years ago, she had NONE of these issues in her classroom. When I was a kid, we all ate PB&J sandwiches on gluten-laden bread with a carton of milk!
I'd say your son shows signs of this, but I would ADD nutrients in a comprehensive manner rather than start time-consuming and restrictive elimination diets. You'd be surprised that you could well eliminate or at least lessen his other problems as well as this diarrhea issue. Diarrhea is a symptom of different types of inflammatory bowel disease, and all of those types are responsive to specific nutritional additions. Another friend's child had SIXTY food allergies - really. Not just the "usual" things like dairy, legumes and tree nuts, but also certain fruits, meats, and many, many other things. He's allergy free - it took about 5 months and she was consistent, but it was easy easy easy because she learned the food science involved.
Let me know if you want more info.