The gravitational redshift as measured in numerous experiments confirms Newton's emission theory of light according to which, in a gravitational field, the speed of light varies just like the speed of ordinary mechanical objects:
http://www.einstein-online.info/spotlights/redshift_white_dwarfs Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."
That is, the shift in frequency is caused by the shift in the speed of light in a gravitational field, as predicted by Newton's emission theory of light. It can be shown that, if the speed of light varies with the gravitational potential as predicted by the emission theory (c'=c(1+gh/c^2)), then, in gravitation-free space, it varies with the speed of the observer again in accordance with the emission theory (c'=c+v):
http://www.youtube.com/watch?v=FJ2SVPahBzg "The light is perceived to be falling in a gravitational field just like a mechanical object would. (...) The change in speed of light with change in height is dc/dh=g/c."
Integrating dc/dh=g/c gives:
c' = c(1 + gh/c^2)
Equivalently, in gravitation-free space where a rocket of length h accelerates with acceleration g, a light signal emitted by the front end will be perceived by an observer at the back end to have a speed:
c' = c(1 + gh/c^2) = c + v
where v is the speed the observer has at the moment of reception of the light relative to the emitter at the moment of emission. Clearly, the speed of light varies with both the gravitational potential and the speed of the observer, just as predicted by Newton's emission theory of light.