universe - Expanding space and red shift

Assuming space is really expanding -

Due to expanding space, distant galaxies are supposed to be moving away from us.

When light leaves a distant galaxy, it's wavelength is red shifted to begin with due to doppler's effect.

But thereafter, the light keeps propagating through space which is continuously expanding. With expanding space, the red shift should keep on increasing for billions of years while it travels.

Therefore, the cumulative red shift must be much larger than what should be due to the doppler's effect at the onset.

Has this effect been taken into account when the calculations of rate of expansion of the universe are done?

Because, farther the source, more time the light has to travel through the expanding space, causing the red shift to be even greater.

Also, isn't it possible that some galaxies may be travelling towards us but because of cumulative red shift (due to expanding space), they appear to be moving away?

Same logic should apply to the gravitational waves, their frequency, wavelength, and amplitude.

Question is - does the cumulative red shift (due to travel through ever expanding space), make any sense? If so, has it been accounted for while doing calculations?

Answer

The redshift of distant galaxies is mainly due to the expansion of space whilst the light has been travelling towards us. The "cumulative redshift" you refer to is the "cosmological redshift".

Galaxies also have a "peculiar" velocity with respect to the co-moving cosmological rest frame. This produces a regular doppler shift.

An example helps. There are 100 galaxies gravitationally bound in a cluster of galaxies that is so far away that the average redshift is 0.1 (the light has wavelengths 10% longer). This redshift is dominated by the cosmological redshift.

However, if the galaxies are taken individually, some have redshifts a little bigger than 0.1, some a little smaller (for a typical cluster, this might amount to a dispersion of 0.002 in redshift). This is because the galaxies have their own peculiar motion with respect to the cluster.

As I say, peculiar motions tend to produce redshifts of magnitude 0.001-0.002. The cosmological redshift increases with distance (Hubble's law!), so once galaxies are far enough away that their cosmological redshifts are larger than this (much more than about 300 million light years), then the peculiar motions become negligible.