Inflation is supposed to explain this: It could provide the initial impulse to kickstart the velocities. I think the general idea is that the fact that everything is moving away from each other to begin with is explained by inflation, and the fact that this expansion is accelerating is explained by dark energy. Take all this with a grain of salt, here we approach the limits of my tenuous understanding, but what I do understand is that none of this is experimentally verified: No “inflaton” has been found, or any other mechanism to otherwise explain inflation theory has ever been produced such that we could test it, and no working model of dark energy has ever been produced (to my limited knowledge) that we could test or detect.
Tl;dr: I’m pretty sure it’s untestable anyway, we basically will never know during our lifetime short of some breakthrough in physics.