For a universe described by the RW metric, a relation between the scale factor at the time of emission of light and the redshift can be derived, and yields
$$
a(t_e) = \frac{1}{1+z}
$$
The above equation depends only on the time of emission and the redshift. The above relation implies that the redshift of any light from any source at any distance is the same, and depends only on the scale factor of the universe at the time of emission. In an isotropic and homogeneous universe (in large scales), the scale factor is only a function of time. How does this settle with Hubble's law, which states that there is a relation between the redshift and the distance between galaxies?
[Physics] Redshift-distance relation, and redshift-scale factor relation
cosmologygeneral-relativityredshiftspace-expansion
Related Solutions
In Stephen Weinberg's well-known book "The First Three Minutes," he talks solely about Doppler shifts. A good review and analysis of the "expanding space" vs. Doppler shift question was provided by Bunn and Hogg, "The kinematic origin of the cosmological red shift," Am. J. Phys. 77 (8), 2009, pp. 688-694. They make convincing arguments that the red shift is best understood as a series of Doppler shifts in overlapping space-time regions small enough so that Minkowski (flat) space-time geometry is an excellent approximation. Regardless, they say in the Conclusion: "There is no “fact of the matter” about the interpretation of the cosmological redshift: what one concludes depends on one’s coordinate system or method of calculation." Their argument for the Doppler shift says it is more "natural" because it is consistent with a number of well-established facts about the general relativistic theory of gravity.
When calculating redshifts, we usually look for signature features in astronomical spectra, usually emission or absorption lines.
For example, the universe contains lots of hydrogen. From quantum mechanics, we know that hydrogen has many different energy states which are fixed. This means it can only emit photons with a particular set of wavelengths (these energy states are like a unique fingerprint for each element). So we know that hydrogen in the distant universe will emit photons with exactly the same wavelengths as we can measure in laboratories on Earth.
Here is a nice cartoon of the redshifting of spectral lines:
You see that the pattern of lines stays the same, they are just shifted to redder (longer) wavelengths.
When light travels through the universe, the wavelengths of the photons are stretched as the universe expands, so the wavelength we measure on Earth $\lambda_{obs}$ will be larger than the original emitted wavelength $\lambda_{em}$ (and we generally know what $\lambda_{em}$ is because it will form part of this 'fingerprint'):
$$\frac{\lambda_{obs}}{\lambda_{em}} = 1 + z = \frac{R_0}{R} $$
The scale factor today $R_0 = 1$, so we can find the redshift $z$ and the scale factor of the universe when the light was emitted $R$.
Best Answer
The short answer is that, as you said, the redshift depends upon the scale factor at the time of transmission (as compared to the present). Since light travels at a finite speed, light from more distant sources was transmitted at a different time and hence scale factor.
You're redshift equation does NOT imply the same redshift for any distance, I think you were just interpreting, forgetting that light we're currently receiving from distant and near (relatively speaking) stars was released at VERY different times (and hence scale factors). The Hubble relation follows directly from the redshift equation for an expanding universe.