Distant galaxies are moving as space expands, not moving through space. So a photon coming to us from them has to come back through distorted space. So is not spacetime distorted, so the photon has to go through an area where time slows down. So would this cause red shift. So maybe we have a simplistic view of red shift and maybe those galaxies are not going as fast as we think
Redshift is the "stretching" of the wavelength of light as it travels, over finite time, towards us, due to the ongoing expansion of space.
The effects of this redshift are indistinguishable from that caused by the doppler effect, so that is why some elementary treatments choose to discuss it in that way.
Most galaxies only have peculiar velocities with respect to thei comoving volume of a few hundred km/s, and this does perturb the cosmological redshift by that (small) amount. That is why Hubble's law only applies at distances such that cosmological expansion dominates.
It sounds like the question is asking if there can be two separate contributions due to the expansion, one simply because the distances are increasing so successive wavefronts in the wave train arrive later and later, and a second effect due to more esoteric distortions like time slowing down. But what of the redshift we attribute to something happening to space, and what we attribute to something happening to time, depend on the coordinates we choose to describe what is going on. The standard choice is comoving coordinates, where time has the meaning of "age of the universe at the point in question in a frame moving with the average gas." In those coordinates, the elapsing time isn't doing anything different as the universe expands, so the only effect you need to worry about is the stretching distances between wavefronts.