I have a diary allergy (not sensitivity and not intolerance - actual allergy to whey and casein - sorry if I'm preaching to the choir) but have not yet had serious issues with cross-contamination. So I asked during Holiday's Around the World 2016 for diary-free options at a few different kiosks - could something be served without sauce or cheese and be dairy-free. I was always able to wait at each kiosk a few minutes for an answer.
If I had ever had significant trouble breathing from cross-contamination, I would not be willing to risk a kiosk. I am fortunate that I get certain sensations that let me know if there was accidental dairy in what I just ate and, so far, benedryl can diminish my reaction to not life-threatening. I prepare the vast majority of my own food and only risk "cross-contamination" on vacation and for less than one serving a day. I would not eat at a kiosk every day...
That's my experience for what it's worth...