I did not know that the idea was at that time already in use in Mongolia. On top of many hills here are piles of rocks that are considered holy, or at least a representative of the spirit of that area. So Mongolians have for years added something to the pile, and walked around it three times. If they don't have a cloth, bottle, food, then a rock or even money will suffice.
Yesterday my neighbor, who is believes in Jesus, stopped by for a visit. I was asking him some questions about Mongolian religion. He said he and his family have never followed many of these customs. Said his dad used to have no qualms about helping himself to visible cash that other people places up there. Said that he himself has never been afraid to pick up bills worth up to 5$ that other people had recently offered to that spirit, mountain, or whatever.
So maybe this helps me answer the question about what I should have done when my dog starts eating the food someone standing right there has just offered to the shrine.