Teaching your horse to lower its head is beneficial to endurance riding in a couple of ways.
Have you ever seen an excited horse with its head down? The dropping of the head releases endorphins which help calm the horse. By using this at your vet checks, you can help lower the heart rate. It can also assist with keeping your horse calm while trying to mount.
It can encourage the horse to drink.
It can discourage the horse from raising its head when it climbs a hill, keeping them more balanced
By helping the horse lower its head, it can better look at an object instead of spooking away from it.
How you can teach your horse to lower its head:
Stand in front of your horse and lower your body, encouraging your horse to come down to your level with their nose. Gently put pressure on the lead rope and release every time for a few seconds when the horse lowers its head in response to the pressure. Eventually, your horse should hold that lowered head position until you release your cue and allow it to raise its head again. You can use this same process when standing beside your horse but instead of using pressure on the lead rope, use your fingers to put slight pressure at the poll and release that pressure every time the horse lowers its head. Once your horse has that down pat, combine that cue with a point on their neck and when you put pressure on the poll also put pressure at that spot on their neck, releasing when the horse drops his head. This way, when you are in the saddle and want your horse to drop its head, you will just have to touch that spot on their neck. The goal is to have your horse keep their head down until you release your cue by removing your hand.
When I’m out riding, I will practice this at puddles I come across. If you have a horse that isn’t the best drinker, this can sometimes help. By just getting their nose down to it, it can sometimes encourage them to drink.