If they aren’t putting us on, they are nuts :)
The dog would be safest running alongside, if not - then teach him to ride behind the rider. Good heavens, they are riding drafts - plenty big enough butt for a dog to sit/lay on.
That is pretty nuts and hey! That dog is a herding dog! He is supposed to be able to herd cows, sheep or what ever. If he can't stay out of the way of livestock feet then he no one can!