I used to have the same problem with my mare. I fixed it by getting her super collected at the trot for a few strides. Then I would literallythrow the reins at her while putting my calf on. IIf she ran off I would squeeze with my thigh. When you start so collected you put yourself in a place to push them up into the bridle.
When you change the length of the horses neck you force them to balance differently. This means if you allow them a longer neck, they are also able to take a bigger step behind. This is why when teaching them to stretch you can throw the rein at them. Once they get better at it you can maintain a soft feel.
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You are thinking VERY backwards! Collection happens from a horse that already understands how to work up into a bridle. Gradually the balance achieved from working into the bridle at the working paces, can be developed into collected paces.
By pulling the horse back into a short trot, and then throwing the reins away, your horse has only two choices. Throw the head into the air to stay in balance, or throw the head towards the ground to stay in balance. Either way the horse is on the forehand and the backs stiffens. The bigger hind legs happen because they have to catch up with the front legs. If you trip while running, and try to save yourself, you naturally start taking longer, exaggerated steps and if you're lucky, you don't fall.
Imagine if we were standing facing each other, holding both hands. If it is my job to hold you in balance, while you lean backwards. You are relying on my to stop you from falling. What happens if I suddenly let go? You end up flat on your back and you're probably not going to trust me near as readily next time we try the exercise. It is no different for a horse.