[Moderator's Note: This thread was split off from a previous thread because of a change of topic.]

It doesn't. Matter influences the curvature of spacetime locally; the curvature of spacetime propagates to distant locations; and the curvature of spacetime at those distant locations tells the matter there how to move. In a static or nearly static situation, the "propagation" part is not easily tested; but there are small effects that show it. One of them is the perihelion precession of the planets; that wouldn't be there if gravity were an "instantaneous" Newtonian force acting through empty space.

But if as you suggest matter influences the curvature of spacetime locally through what mechanism does it achieve this? A billiard ball moves along a flat table as a result of a collision by the cue transferring its momentum to the ball; after it has left the cue it cannot be persuaded to return to the cue without a change of momentum occurring in the opposite direction. If you put the ball inside a straight tube it wouldn't move; if you put the ball inside a curved tube it still wouldn't move. How can space which is essentially void have a shape? What actually causes a particle of matter to move from location A to location B in the vacuum of space within a gravitational field?

The Einstein Field Equation describes how it happens. I don't know what exactly you would consider a "mechanism", but apparently you mean something like "anything that looks like a mechanism I would see in everyday life" (see below). That's not a good heuristic.

Is this what you mean by a "mechanism"? Do you understand how much complexity underlies this process?

So what?

You mean, it wouldn't move if no other interactions were present and the tube were horizontal (or floating in free space, far from all other objects). Again, so what?

GR doesn't say gravity is a manifestation of space curvature; it says it's a manifestation of spacetime curvature. More precisely, tidal gravity is a manifestation of spacetime curvature; in fact it is spacetime curvature. So if the term "spacetime curvature" bothers you, just substitute "tidal gravity". You'll preserve all the correct predictions of GR.

The same thing that causes it to move from location to location in the vacuum of space without a gravitational field. Freely falling objects are the default state, so to speak: they just "move through spacetime" following whatever the natural pathway is in their local region.

Peter is correct. But you seem to searching for an answer to a very deep question! Your question is basically: How does action at a distance work? The truthful answer (all virtual particles aside) is: we don't know - it just does.