Being equivalent, the gravitational frequency shift and the Doppler frequency shift (moving observer in gravitation-free space) are often deduced together:
http://galileo.phys.virginia.edu/classes/252/general_relativity.html Michael Fowler, University of Virginia: "What happens if we shine the pulse of light vertically down inside a freely falling elevator, from a laser in the center of the ceiling to a point in the center of the floor? Let us suppose the flash of light leaves the ceiling at the instant the elevator is released into free fall. If the elevator has height h, it takes time h/c to reach the floor. This means the floor is moving downwards at speed gh/c when the light hits. Question: Will an observer on the floor of the elevator see the light as Doppler shifted? The answer has to be no, because inside the elevator, by the Equivalence Principle, conditions are identical to those in an inertial frame with no fields present. There is nothing to change the frequency of the light. This implies, however, that to an outside observer, stationary in the earth's gravitational field, the frequency of the light will change. This is because he will agree with the elevator observer on what was the initial frequency f of the light as it left the laser in the ceiling (the elevator was at rest relative to the earth at that moment) so if the elevator operator maintains the light had the same frequency f as it hit the elevator floor, which is moving at gh/c relative to the earth at that instant, the earth observer will say the light has frequency f(1+v/c) = f(1+gh/c^2), using the Doppler formula for very low speeds."
That is, the frequency as measured by the "earth observer" is
f' = f(1 + gh/c^2),
and the frequency as measured by an observer moving with speed v towards the light source, in gravitation-free space, is
f' = f(1 + v/c).
If, in a gravitational field, the speed of light "varies with position", as Einstein claims, then the speed of the light relative to the "earth observer" is
c' = c(1 + gh/c^2),
and the speed of the light relative to an observer moving with speed v towards the light source, in gravitation-free space, is