A low headset is trained. For my horse it was anyways..It took many months of consistensy and training. What my trainer told me to do to teach the horse that they will be happier and more comfortable if they just do what you ask. We started off at just standing still getting her to lower hear head, then moved up to walking, then trotting, and were still working on cantering. But the most advice I can give you is to be consisten and don't give in!
Also, to get the horse to put its head where you want, I would use the "leg, leg, head down" method where you pull, not hard, but gently, one rein to one leg, then the other rein to the other leg and ask for the head to be dropped. This can be done at all speeds and its SO helpful.