If interpreted correctly, the Doppler effect directly refutes the postulated independence from "the state of motion of the emitting body". Here is an incorrect interpretation - the postulate is saved by wrongly assuming that the light pulses bunch up in front of the moving source:
Albert Einstein Institute: "We will start with a very simple set-up, which you can see in the following animation. On the right-hand side, drawn in green, there is a sender that emits pulses in regular succession. On the left-hand side there is a receiver, drawn in blue. The pulses themselves are drawn in red, and they all travel at the same speed from right to left. Everytime the sender emits a new pulse, a yellow indicator light flashes once. Likewise, a flashing light indicates when a pulse has reached the receiver:
Next, let us look at a slightly different situation, where the source is moving towards the detector. We assume that the motion of the sender does not influence the speed at which the pulses travel, and that the pulses are sent with the same frequency as before. Still, as we can see in the following animation, the motion influences the pulse pattern:
The distance between successive pulses is now smaller than when both sender and receiver were at rest. Consequently, the pulses arrive at the receiver in quicker succession. If we compare the rates at which the indicator lights at the receiver and at the sender are flashing, we find that the indicator light at the receiver is flashing faster." [END OF QUOTATION] http://www.einstein-online.info/spotlights/doppler
Einsteinians make the following assumption above, which is essentially identical to Einstein's 1905 constant-speed-of-light postulate:
Assumption 1: "The motion of the sender does not influence the speed at which the pulses travel."
Assumption 1 goes hand in hand with another assumption:
Assumption 2: "The distance between successive pulses is now smaller than when both sender and receiver were at rest."
Assumption 2 is false - the pulses do not bunch up when the source (sender) is moving. If they did, by measuring the (variable) distance between the pulses, an observer associated with the source would know whether he is moving or at rest, which contradicts the principle of relativity.
Since Assumption 2 is false, Assumption 1 is false as well. If the speed of the moving source is v, the speed of the light relative to the receiver is c'=c+v, in violation of Einstein's relativity.
The following quotations suggest that, if the speed of light is variable, modern physics is dead:
"The speaker Joao Magueijo, is a Reader in Theoretical Physics at Imperial College, London and author of Faster Than the Speed of Light: The Story of a Scientific Speculation. He opened by explaining how Einstein's theory of relativity is the foundation of every other theory in modern physics and that the assumption that the speed of light is constant is the foundation of that theory. Thus a constant speed of light is embedded in all of modern physics and to propose a varying speed of light (VSL) is worse than swearing! It is like proposing a language without vowels." http://www.thegreatdebate.org.uk/VSLRevPrnt.html
"But the researchers said they spent a lot of time working on a theory that wouldn't destabilise our understanding of physics. "The whole of physics is predicated on the constancy of the speed of light," Joao Magueijo told Motherboard. "So we had to find ways to change the speed of light without wrecking the whole thing too much." http://www.telegraph.co.uk/technology/2016/12/06/speed-light-discovered/