Some observations:
FP requests may come at any stage of the process right up to the day the oath is taken.
Most reports, however, are for requests made early in the process, similar to the timing reported by Adonis_G, as in soon after AOR. This is of course related to the most common reasons for the FP request, which are related to verifying the identity of the applicant -- recognizing that concern about the applicant's identity can arise for different reasons, ranging from distinguishing clients with similar names and DoB to distinguishing the particular individual in RCMP name-record hits, to direct concern about identity arising otherwise, such as due to imprecise or difficult to verify information in the applicant's passport(s) or travel document(s), such as confusion arising from or related to name issues.
In the majority of cases, FP requests are nothing to worry about and do not cause lengthy delays. What triggers the request may be incidental and largely unrelated to the applicant (but for the coinciding aspect triggering the need to distinguish one person from others). The impact on processing time appears to be relatively minimal most of the time, up to a couple or few months for some.
That noted, anyone with law enforcement contact, and particularly anyone who has been charged with an offence, even if dismissed or otherwise not constituting a prohibition, and including persons who have been identified as suspects or persons-of-interest in law enforcement investigations, is much more likely to be issued a FP request. The FPs can be used not only to verify identity, but also in a broader RCMP or CSIS background check into databases which can connect FPs to persons with criminal history or even to persons suspected in particular crimes or persons believed to have association with criminal or terrorist organizations, formal or otherwise (such as known gangs, Canadian and foreign). Anyone at risk for this should already know this risk exists; a person not only knows whether they have been charged with a crime, but generally knows whether or not he or she has been associating with others involved in criminal activity.
Obviously, if the FP request is related to particularized concerns about prohibitions, or criminality or security issues, there is the prospect of more lengthy delays due to non-routine processing and potential referrals or investigations.
The later in processing the FP request is made, the higher the odds it derives from a concern more particular to the individual. Nonetheless, most FP requests even later in the process are still, probably, incidental, nothing to worry about, still about verifying the applicant is not someone else. And again, if there is something to worry about, the individual almost always knows there is something to worry about (and most protests to the contrary lack credibility)
