PLL for Phase Demodulation and Carrier Tracking
Yes this is very common to have a dynamic loop bandwidth such that during acquisition the loop bandwidth is wider, and then once acquired to tighten it up for better noise performance. 

A typical loop will have an error signal determined which is presented to the input of the loop filter. The filtered version of this error signal can be thresholded and used as a lock determination metric. The loop must also be designed such that the loop bandwidth can be transitioned from wide to narrow without disrupting the lock state. 

Note that even after acquisition you would like the loop bandwidth to be as wide as possible depending on the dynamics of the link (you mention Doppler, of interest after acquisition would be the maximum possible rate of change of the Doppler offset), but you don't want it so wide that it starts to track out the phase variation inherent in the modulation itself. The phase noise of your local oscillator (LO) plays a big part too in optimizing this loop bandwidth for carrier tracking; if you make the loop bandwidth too tight then your LO and similar jitter sources (ADC clock) start contribution to your noise in detrimental ways.

To this point consider the plot below showing the effect of local oscillator phase noise on a QAM signal and how it interacts with a carrier recovery loop:



Consider two "slices" from the phase noise spectral density for an example oscillator:



The carrier tracking loop is a high pass filter to these noise contributions; the the lower frequency phase noise contribution (which has a significantly higher noise level if not filtered) is significantly effected by the carrier tracking loop bandwidth. 



Here is the result of a simulation I had done where you can see how the phase noise contribution is reduced from the rejection provided by the (higher) carrier tracking loop bandwidth.



I wanted to hammer this point home since your question implied that a lower loop bandwidth is a lower noise result and it's actually a minimization problem between LO phase noise/ clock jitter, and carrier loop tracking noise (and signal removal from tracking out the signal).

I go into more detail of this at this post: 

Loop bandwidth for symbol timing recovery