Any redshift/blueshift, in particular the Hubble redshift, is a direct consequence of two foundational principles in future, Einstein-free physics (the second principle is a corollary of the first):
1. The wavelength of light is invariable.
2. Any frequency shift is caused by a proportional speed-of-light shift.
The same holds true for the gravitational redshift/blueshift:
University of Illinois at Urbana-Champaign: "Consider a falling object. Its speed increases as it is falling. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth. Because of the equivalence between gravitational and inertial mass, we should observe the same effect for light. So lets shine a light beam from the top of a very tall building. If we can measure the frequency shift as the light beam descends the building, we should be able to discern how gravity affects a falling light beam. This was done by Pound and Rebka in 1960. They shone a light from the top of the Jefferson tower at Harvard and measured the frequency shift. The frequency shift was tiny but in agreement with the theoretical prediction."
https://courses.physics.illinois.edu/phys419/sp2011/lectures/Lecture13/L13r.html
The same holds true for the Doppler redshift/blueshift:
"Thus, the moving observer sees a wave possessing the same wavelength [...] but a different frequency [...] to that seen by the stationary observer."
http://farside.ph.utexas.edu/teaching/315/Waveshtml/node41.html
"Let's say you, the observer, now move toward the source with velocity Vo. You encounter more waves per unit time than you did before. Relative to you, the waves travel at a higher speed: V' = V+Vo. The frequency of the waves you detect is higher, and is given by: f' = V'/λ = (V+Vo)/λ."
http://physics.bu.edu/~redner/211-sp06/class19/class19_doppler.html
"Vo is the velocity of an observer moving towards the source. This velocity is independent of the motion of the source. Hence, the velocity of waves relative to the observer is c + Vo. [...] The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time."
http://a-levelphysicstutor.com/wav-doppler.php
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