I taught my colt to lay down on command. And honestly you can't just follow the horse around and when you see them lay down. Say down and praise them. Or they'll never learn. When you first teach a horse it looks quite cruel, until you can finally use no ropes to lay them down. What you do is stand on the left side of the horse. And with a halter on and a lead rope attached pull their head to the right by pulling the leadrope under them from the left side. While pulling the left leg slightly back. And eventually he will lay down. And immediately have the treat ready. At first they may go down hard. And it may take a struggle to finally show them the way. But eventually they will go down softer. I have sparta where I can use just a neck rope and pull his leg and he will lay down nice and soft all the way onto his side and then look at me like "treat!?" and then while they are down just give him alot of rubbing and love and then cue your command for them to get up. I think it took sparta at least 5 times of hard tries to get him to understand exactly what I wanted.
Baby, Sparta, Carmen, Henry, Hooch, Mercedes, Butterscotch