I know it is a strange concept for human's to accept but we really are not moving ... except with respect to the other members of our local group of galaxies. At least not as defined by astrophysics.

What is happening is that the spacetime between our local group and other groups of galaxies is increasing. We measure a change in distance. But there really is no movement as there is no outside frame of reference from which movement could be measured.

Hope I didn't confuse things further.

DA Morgan