Consider a cloud of particles falling into a black hole. How does the relative velocity between two such particles, one which is already at the event horizon and one that is still some distance away from the event horizon, depend on the distance between them, as considered in the reference frame of the particle that is still some distance away from the event horizon?
Based on the fact the nobody even tried to answer this question, I'll venture to answer it myself. I think I understand it well enough at this point to do so, but I'm not perfectly sure.
The relative velocity between a particle approaching the event horizon and one that is at the event horizon in the reference frame of the former is the speed of light. This relative velocity is valid regardless of the separation in time or space between the two particles, provided that we consider this separation in the reference frame of the former particle. In other words, even if the first particle is only a fraction of an atto-second away from crossing the event horizon, the difference in velocity between itself and the particle at the event horizon is the speed of light.