Perhaps the answer lies in relativity itself. The measurements seem to show that the farthest objects that can be seen in the universe are accelerating away from us. But, since motion is relative, it also means that we are also accelerating in our motion through spacetime. We cannot measure our own motion so easily as we can very distant objects, so we need to use indirect inferrence.
By the way, it is said that we should regard the expansion rate of the universe as generating true kinematic motion, not just a "stretching" of spacetime. The idea here is that we shall avoid confusion this way. So, it is perfectly proper to speak of the expansion of the universe as if it gave rise to actual motion.