I have over the years had two nice hunters that were "head shakers" They flip their nose suddenly, quickly up and down, often very quickly. One even went so far as to lift his front leg up when he flipped his nose up. There have been several ideas, treatments, issues "tossed" around throught the years. Some horses tend to do it more when first worked in bright sunlight. Some it is seasonal, some it is regional, some you just never know. we first flush the tear ducts to be sure they haven't gotten a blocked tear duct causing pressure in the nose. The duct drains at the end of each nostril. We then check allergies, scope and X-ray head. All have been within normal limits. Things that have been of great help.
Antihistimine treatment, Cyproheptadine, 3 day withdrawl to show. Cover the entire nose with a thick riders hair net. Now they actually make nets to cover the nose Dover Catalog has them for horses now, we used to just use the thicker nets you buy in tack stores. We lightly spritz the end of each nostril with the medication used when they intubate people. When you are standing around between classes we drape a damp wash cloth over the horses noseband, long enough to cover the nose. I have found a softer feel, even a "sidepull" helps a little. Riding the horse on the bit makes no difference, in fact I have seen the condition get worse. One of the horses almost completely recovered once we moved back to Oklahoma, only at show where they used a lot of Cedar to decorate jumps, did he start to revert. Nothing is more distracting than riding down to a 3 foot 6 inch line and have your horse disrupt the stride and distance by flinging their nose straight up. Good Luck