In theory, a running martingale should provide a place with a release from the pressure if the horse keeps his head below a certain point.
What I dont' like about them it that it takes what should be a direct line of contact and feel from the elbow through the hand to the bit, and puts a "kink" in it, when activated( a downward pull literally puts an angle in what was a straigtht rein to the bit) . It translated the pressure from a straight back and very adjustible one, to one that pulls downward on the bars of the horse.
This is much more uncomfortable for the horse and builds tension in the horse, expressed in the jaw and neck. Horses that spend much of their ridden time in running martingales (or draw reins) often have enlarged muscles on the front of the neck, and this is from them bracing to protect themselves. It becomes a rigid way of holding themselves and can translate into a very hollowed out and stiff back, with the back muscles all firing. This is just the opposite of what the OP is striving for.
You may say, well, the martingale doesn't come into play unless the horse raises his head. True, but if it is adjusted where it comes into play soon enough to teach somethign, then the contact on the bit will go from direct (hand/elbow/straight line = good) to (downward on bars = bad) very quickly, and then back off, on, then off, on then off as the horse moves around trying to find release from this> If the rider does not have very good feel and ability to follow the horse , the connection gets so jumbled that it makes for very bad communication.
If you use the martingale much looser, then it doesn't even come into play until the horse's head is way up. This might be the lesser of the two evils. But, then, why have it? If the hrose throws it's head up enough to break your nose, put a standing martingale on it. Otherwise, if it puts it's head up, you DON"T try to pull it back down by lowering your hands. NO
You follow it up. With your hands, keeping a steady but unrewarding contact on it, until . . . It looks for a way out from the contact by lowering it's head. And I don't mean a total lowering, I mean it flexes a bit at the poll andjust even THINKS about bringing its' head down. THEN you release and reward. YOU TEACH it to bring it's head down using YOUR hands and your unwarped direct line contact to its' mouth.