In arbitrary spacetimes, redshift has to be computed along specific worldlines, ie you tend to treat photons as point particles. I guess in principle, you could

In arbitrary spacetimes, redshift has to be computed along specific worldlines, ie you tend to treat photons as point particles. I guess in principle, you could solve the Einstein-Maxwell equations (or more likely, the Maxwell equations on a fixed background), but I wouldn't recommend it. There are special cases which are simpler to handle (eg if your spacetime is static and you use adapted coordinates, the time coordinate will be cyclic in the particle Lagrangian, the canonically conjugate energy will be conserved and redshift will be given by the quotient of the roots of the g_{00} components, which formally comes in via normalizing emitter and observer 4-velocities). I'm not sure if you are objecting to approaches like the tetrad formalism in your final sentence The final sentence was just a comment on the pedagogy of deriving cosmological redshift via infinitesimal Doppler shifts. There's nothing wrong with it (sections of the frame bundles and all that jazz if you want to get formal), but to me, this feels like putting things on their head: I consider the Doppler shift a special case of the generic relativistic shift, making it circular to then explain the generic case as a series of infinitesimal Doppler shifts integrated along a photon's worldline.