ForThe frequencies corresponding to the casedetermination of range are typically computed using a "range FFT" which is an FFT of the IF signal over a unit time interval (which is typically the FFT chirp duration but need not be), resulting in complex values corresponding to each bin in the FFT and called "range bins".
Velocity of targets is determined by comparing successive range FFT's over a fixed time interval (typically the time between chirps if the range FFT is done on a complete chirp and the chirps are transmitted at consistent time intervals). This can be done with a digital differentiator on the computed ranges if a shorter time velocity resolution is needed, or with subsequent "Doppler FFT's" across multiple range bins: This would involve fast time "Range FFT's" done to compute the range bins (as a column) on finer scale time blocks (such as a chirp) and then with the use of multiple consecutive Range FFT's over a longer time duration (such as over multiple chirps) slow time "Doppler FFT's" are computed across the rows to compute the average velocities. The reason this works is to consider how a slower velocity would change the range slightly from one Range FFT to the next. This results in a change in phase for the particular FFT bin that was maximized for the given range. A change in phase over a change in time (from one Range FFT to the next) is frequency, since instantaneous frequency is $d\phi dt$, so therefore any frequencies across the rows would be a result of changing ranges. The resulting grid of bins as range in columns and velocities in rows is referred to as a "Range-Doppler Map".
The actual frequencies measured corresponding to the ranges in the range bins are modified due to the Doppler offset for moving targets. This has the effect of shifting all the frequencies for the range bins up and down, proportional to the velocity for a given target but does not change (for the case of constant velocity) the change in range over time, meaning the Doppler bins and resulting computed velocities are not modified. This means if we want to do Doppler correction, we can compute the velocities first from the rows in the Range-Doppler map and then with that correct for the Doppler offsets induced on the ranges resolved from the columns.
To see how Doppler offset affects the determined range, consider first the target with a constant velocity moving toward the trasmittertransmitter. Assuming the frequency change over the ramp is significantly smaller than the carrier frequency transmitted (as would be typical for FMCW), the Doppler offset in this case would be constant and independent of the frequency ramp rate, shifting the frequency of the received signal higher as depicted in the graphic below and given by the formula below that.
[NOTE: I derived the above result while writing this post and did not yet confirm with a reference, simulation or other source. There could easily be sign or factor of 2 errors so please confirm my derivation carefully. I will remove this note once I have otherwise validated the result]
A useful reference on FMCW and other radars is this write-up by Infineon.
