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When dealing with the speed of planets and galaxies relative to each other, simple Newtonian math works just fine.

Also, I agree that they are using some fuzzy math here. Supposedly the relative speeds of the galaxies are MUCH lower than c. If the galaxy they are talking about is 12 billion light years away, and the universe is 13 billion years old, then at some point nearly every object in the universe was traveling at or near the speed of light.

Also, using their argument that the light from the mentioned galaxy has traveled 12 billion years to get here, the simple fact that you can see it at all suggest that the relative speed of the galaxy 12 billion years ago was well within the Netwtonian physics range. Otherwise, the Doppler shift would have made the frequency of the light coming from the galaxy so low that it simply could not be seen.

Using those two points of logic assume the earth and the galaxy are relatively the same distance from the center of the "big bang" (6 billion light years) and that their speed is relatively low. That means that they traveled 6 billion light years each within the span of 1 billion years and thus violated the theory of relativity which states that no object can achieve the speed of light except light itself.