In special relativity co-moving objects see the other's 4-velocity as being only temporal.

When they move relative to each other they see the other's 4-velocity has rotated so that it points less in the temporal direction but now has a spatial component.

By the equivalence principle two co-moving objects falling toward a planet see each other's 4-velocity as only temporal in their own (falling) rest frame so they must think the space between them is attached to their rest frame over time. It thus seems that space has the same rotation of its own 4-vector over time (up to a constant if the objects started with a fixed velocity before falling) But surely space does not fall. Also if space-time curvature causes objects to fall, how? I'd have thought it's just a map of how objects move. not a cause of that motion, but if it does cause falling , how?

Space isn't moving so as to push or rotate mater. Surely it's curvature it's just a map of the rotations in (light and) matter's 4-vector? How does something about the mass energy tensor alter geodesics or 4-velocity vectors? I see no explanation of gravity in GR merely a more detailed description of the motions it effects.