Space-Time is our current model for.. well.. space. It is non-Euclidean, in that it curves. Gravity (such as in your black hole example, or for another body such as a planet) is the bending of space-time. As such, you're asking if gravity affects our movement (which it clearly does), not space-time. Asking the question "does space-time affect our movement" is analogous to asking "does the ruler affect the length?" The most honest answer to that question is "we better hope not." The [I]shape[/I] of space-time can affect [I]apparent[/I] motion though. For instance, in our current model, photons (light) do not undergo gravitational interactions. Yet, light "lenses" around planets implying that the photons are in fact being pulled on. The actual explanation is that gravity is bending the space that the photons move through, thus making their path appear curved to us, when in actuality the photon is still traveling in a straight line [I]through space.[/I]