Is it accurate to say space moves because spacetime is curved? I've learned from PBS Space Time videos that space rushes into a black hole's event horizon at the speed of light.  I'm trying to square this with the analogy of stretched rubber sheet everyone uses; in that picture, the rubber doesn't move laterally.  I'm wondering if maybe the rubber sheet actually represents spacetime, which doesn't "move" because time is just another dimension.  If spacetime is curved in the time direction, maybe that appears to us as space moving over time.
Am I on the right track here, or way off base?
Edit
Thanks for the link to the question "Can gravity be interpreted as the acceleration of spacetime towards an object?"  However, I believe that is different.  Spacetime includes time, and therefore cannot be said to evolve over time.  I'm not asking if spacetime accelerates; I'm asking about whether the space has velocity due to the curvature of spacetime.
 A: No, this is not accurate. It is a pictures even used by cosmologists in popular science literature. It is analogous to think of corks in a river, showing it's speed. This picture attempts to give a reason, why objects are crossing the event horizon of a black hole with light speed. But it fails, because in General Relativity space being empty doesn't flow and and doesn't take things with it like water does due to it's viscosity. Instead spacetime tells things how to move, in this case across the horizon.
The rubber sheet analogy shows that things move away from each other, more precisely accelerated to each other in curved spacetime. If a rubberband falls radially towards a black hole it get's stretched more and more. In this sense it demonstrates how tidal forces act. A rubber sheet get's stretched too in two directions in an accelerated expanding universe.
So tidal gravity does not correspond to the velocity of a freely falling object relativ to a shell observer (observer at $r = const.$). This velocity can be extracted from the Schwarzschild metric $dr/dt = -\sqrt(r_s/r)$. S0 crossing the event horizons means $ r_s = r$ and hence the velocity $c$ with $c = 1$. The sign of the velocity is negative, because the object moves away fromn the shell observer. Note there is no real observer at $r = r_s$ because nothing with restmass can hover there.   
