There is a non-linear effect of Loop BW on Capture range and lock-in time.
THe error signal in f must be within the loop bandwidth.
Using CMOS analog switches , one can change the bandwidth after locked as long as an offset is not induced or a large transient pF pulse discharge. Or one can use a linear AGC approach to change bandwidth smoothly.
All mixers are phase detectors and all phase detectors are mixers unless phase/frequency type II digital mixers, which have full capture range.
Down side is they have more phase noise locked on rising edge and some dead-time, instead of averaging types that lock on at 90deg, but total lock-in range.
Gain affects stability, phase margin and capture time and a lag-lead filter is essential C1//R+C2 where C1/C2 ~10:1 typ.