I can’t speak to tradition but in the August 2019 feature on horchata in the LA Times they found dairy in a few well known Mexican spots in LA.
“Does your horchata have dairy in it?” I asked after seeing a notice on the menu at the West Hollywood location of Guisados. The cashier said yes. A couple days later at La Playita Siete Mares in Silver Lake, I asked the same question and got the same answer. And then the same at Tacos 1986 in downtown L.A. After that third time, I started questioning whether I had been mistaken about my favorite agua fresca.