The moon is, on average, about 384,000 km away from you. Your motion on the earth, whether it is a few meters, or even several kilometers, will not change the moons position in your sky very much.
Think about it this way.
When you are driving in a car, and look out the window, the nearby trees seem to zip by you very fast. More distant lightposts zip by more slowly. And the mountain range far off in the distance hardly seems to move at all over the course of a couple of minutes. Now, the moon, being much, much more distant than any of these objects, is not going to change its position very much relative to you, as the change in angle is completely negligible.