If the speed of light is a constant, the Doppler Effect in light cannot exist or the need arises for a completely different explanation of how the Doppler Effect works in light. It must be arbitrarily constructed simply to support assumptions of relativity.

A better understanding of  the underlying cause of frequency is one of the prerequisites for a deep understanding of the Doppler Effect. To put it in simplistic terms, one of the causes of frequency is the relationship between the initial force applied to the medium and the elasticity, or 'reactiveness' of the medium, the other is just the speed at which the emitter itself is vibrating. This difference could be exemplified in the difference between a tuning fork and a firecracker. The tuning fork's vibration is faithfully transmitted into the surrounding air whereas the sound of a firecracker is caused entirely by the compression of the media. Amplitude is also involved with compressing the media because the more force applied to the media before it can react, the higher the amplitude will be.

Below is a visual representation of frequency to establish a baseline for the following visual explanations of the Doppler Effect. What you see here is the relation between frequency and wavelength.



Below we have an example of Doppler Effect caused by a stationary sound source and a moving listener. (Note that stationary vs moving is determined by the medium)
In this example, though the sound was inscribed on the medium at a frequency of 100, the listener was moving so therefore the wave appears to be a shorter wavelength (higher frequency) than we know it to be because our perspective is the same as the medium. The listener simply traversed more waves during the measuring period than he would have if he remained still.

Note that if the moving listener above measured the speed of sound from his perspective it would be measured as 680.58 m/s.


In this example we have the Doppler Effect caused by a moving emitter (in relation to the medium). In this example, the medium itself is inscribed with a higher frequency sound. What is happening is that the speed at which a single wavefront can escape from the emitter is slower now in comparison to the Jet's (emitter's) speed. A single wave is only able to travel half as far away from the emitter before another wave is inscribed into the medium. Therefore the wavelength is half as long, the frequency is increased.



In the final example above you can see that the speed of the emitter has overcome the ability of the medium to transmit the sound. The jet passes right through a wavefront as it inscribes it. The compressed area just can't get out of the way fast enough. What you can see is that the speed of sound is actually a property of the medium. It describes its elasticity. It describes how quickly compressed air can normalize itself back to its original pressure by re-distributing the pressure into the surrounding air.

Note that the final example is just a thought experiment because the air is so disturbed around something traveling that speed that there are many more considerations than we need to get into.

Summary

Doppler, red/blue shift, frequency change, and wavelength change -whatever- caused by motion happens by one of two mechanisms:

1) The listener traverses additional waves during the test period because the speed of sound relative to the listener is greater.

2) The individual waves escaping from the emitter cannot get as far away from the emitter before the next wave is produced because of the difference between the emitter's speed and the speed of sound. (elasticity/reaction speed of the medium) The speed of sound is measured to be the same to the listener in this case though.


Without a medium and with the speed of light always constant, neither of these conditions can be met to cause a reduction or increase in wavelength in light. The only explanations of light Doppler cannot go to the same depth of explanation. The simplistic notion that the wave has more or less energy based on the shift is severely lacking depth. It is a description instead of an explanation. The actual mechanism of the wavelength change cannot be explained in this illogical view.