The fact that photons can’t speed up or slow down explains why light escaping massive bodies turn red. It also explains why bodies moving away from us appear redder than bodies moving towards us.
The way to think about this is to first consider ordinary matter, and then apply what we know to be true for such bodies to light.
In cases where ordinary matter would absorb energy by speeding up, light will turn bluer. In cases where ordinary matter would loose energy and slow down, light will turn redder.
Since a ball tossed towards us from an oncoming object will move faster than a ball tossed towards us from a receding body, we know that photons reflected by a body approaching us will be bluer than photons reflected by a receding body.
Keep in mind that all photons can be considered to be reflected, also those that are excited by a lamp or radio transmitter. Photons originate in the aether, and are therefore reflected by bodies, regardless of whether they are exited or not.
Note also that it doesn’t matter whether it’s the observer or the observed object that moves.
If the observer travels towards an object, the photons will be registered as bluer than if the observer travels away from the object.
This is because the observer adds energy to the interaction with the approaching photon when moving towards it. Conversely, the observer subtracts energy from the interaction when moving away from a photon.
Red-shift and blue-shift of light is therefore entirely about relative motion. It doesn’t matter what energy level the photon has while traversing the space between the observer and the observed. All that matters is relative motion. If the observer and the observed travel at identical speeds, no net red-shift will be detected, even if both travel at a tremendous speed relative to some other object.