Let's assume two observers $A$ and $B$ hovering in a gravitation field. $A$ sends a radio transmission of frequency $f_1$ to $B$. $B$ receives this transmission and finds it has frequency $f_2$.
As as second experiment $A$ sends an electron beam to $B$. They measure the energies of these electrons on emission and reception.
Particles have de Broglie frequency which is proportional to energy. Will this frequency gets redshifted the same way as photon frequencies in a gravitational field so the rate of the original and redshift frequency/energy will be the same as in the case of photons?
In other words, for example, if a 900 keV photon, fired from $A$, gets red shifted to 850 keV when it arrives at $B$, will any massive 900 keV particle get slowed down to 850 keV after it free falls to $B$? - Given the rest mass is smaller than 850 keV, otherwise it would just fall back and never reach the other observer I guess.
I worked it out in flat spacetime between two accelerating observers that keep fixed distance between them, and $A$ just drops particle and $B$ just catches it. And in that case it seems the rate of the total energy of the received and dropped particle is exactly the rate of acceleration induced time dilation between the two observers. I'm unsure if this only works this way in this specific case or if it works in general in any gravitational or fictitious force fields.