A falling body carries with it whatever speed it had originally PLUS what ever speed it picks up falling through into the gravity well of the other body. As it falls, the other body also falls towards it, in such a way that the gained momentum of body 1 is exactly equal and opposite the momentum of body 2 before and after the collision. Which is 0. In other words, M1V1 = -M2V2 = 0. The velocity the bodies gain falling towards each other cancel out.

However, whatever speed the two bodies had relative to each other prior to their interaction could be used to increase the momentum of the entire system of mass M1+M2. So if two bodies are traveling through space with a relative velocity of v prior to the collision, the momentum of the combined system after an elastic collision is (M1+M2)v. However, any momentum gained by the two gravitationally falling towards each other is cancelled out. If this is the case however, the velocity of the two bodies prior to collision must have been enormous, which does not answer the question of where does that velocity come from. It doesn't come from potential energy being converted to kinetic.

I don't think these high stellar velocities are a result of momentum transfer. They are either caused by stars of the halo population zipping through the disk orthogonally and not moving with the general drift of the local standard of rest, or they are members of old binary pairs which were flung off when their primary star went supernova.