1. Distant galaxies are red-shifted.
2. If the red shift is due to Doppler effect then everything is moving away from the Milky Way, which is the center of the galaxy.
3. If something is moving away from the Milky Way, it will be red-shifted due to the Doppler effect.
4. Hubble discovered metric expansion of space as an alternative explanation for the red-shift.
5. We are now certain that the universe is undergoing metric expansion because of the discovery of CMBR.
6. We believe the expansion of the universe is causing cosmological red-shift which has nothing to do with motion.
The expansion of the universe has zero effect on matter. It does not cause matter to move, it does not accelerate matter, it exerts no force, does no work, and there is no energy associated with the expansion of the universe. The cause of expansion operates on space and has no direct effect on matter. Gravity cannot counteract expansion because gravity moves matter through space and expansion creates space itself.