What is space 'doing' near the event horizon of a black hole?

So far as local physics is concerned, that is what a freely-falling observer it depends on what kind of black hole it is...whether it is rotating or not.

The details are very complicated, but the common feature is that nothing peculiar really happens. You do not feel a 'lurch' or 'space discontinuity' or something. The gravitational tidal force will continue to grow as you near the horizon and pass through it. Your clocks and meter sticks will still look normal if they are not too big compared to the tidal field. Just outside the horizon, photons will be able to go into orbit at a 'last-stable' orbit radius. Don't worry about what the distant observers see. That's completely irrelevant to what you will be experiencing locally.

Another thing you will discover as you approach the event horizon is that you are no longer able to go into a stable orbit. Because of a relativistic phenomenon known as the Lenz-Thirring Effect or 'frame dragging', space-time is no longer a static thing, but becomes very obviously dynamic in character. It's a bizarre process that basically causes space-time to be dragged along with the rotation of the black hole so that you can never go into an orbit that is close to the horizon. All you can do is be 'dragged' across the event horizon. This exotic process has been detected in careful X-ray studies of nearby black holes as we witness the spiraling-in of matter.