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I think most of the burning/unburnt powder remains in the case/chamber until after the bullet exits the muzzle, at which point the rapid decompression pulls it forward.
There's really no reason why the powder would go flying down the bore during the initial burn. Some of it might, but it's not going to be pushed the same way the bullet is, pressure can get past a piece of powder, it can't get past the bullet. The building pressures would be more or less equal from all directions on a single powder grain. It started stationary, the only source of movement was the primer blast and the bullet is essentially still stationary when that event ends so the powder has no where to go.
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Except that a deflagration event has a wave front. You are starting a pressure wave with the primer. I don't disagree that unburnt powder in the chamber will move forward as pressure equalizes. However given the temperature difference between front and rear ignition cases that there is significant amounts of powder not burned in the case.
Jimro