Photons are always moving at the speed of light. The speed of matter unless also moving close to the speed of light would not matter (no pun intended). I'm not sure if matter moving at relativistic speeds has an impact on the Compton equations (again no pun intended).

Also as noted above because the deflection would not be uniform this would mean that he spectra would be blurred. Rather then just shifted to the red. The lines would be fatter and dimmer because of Compton scattering. Refraction in the atmosphere has a uniform angle change. Also some of the spectra would be absorbed depending on the matter encountered (this is how the Greenhouse effect works).

Also matter at all times is moving and random in motion unless it has a temperature of 0 Kelvin. This is lower then the background radiation of the observed Universe (something like 3 Kelvin). As long as matter has a temperature above 0 kelvin then matter moves, that is what heat is.