Following up large numbers of candidates in continuous gravitational wave searches presents a challenge, particularly in regard to computational power and the time required to manually scrutinize each of the candidates. It is important to design and test good follow-up procedures that are safe (i.e., minimize false dismissals) and computationally efficient across many search configurations. We investigate two follow-up procedures, or "vetoes," both of which exploit the Doppler modulation predicted in astrophysical signals. In particular, we introduce the concept of using an effective point spread function as part of our veto criteria. We take advantage of a well-established semicoherent search algorithm based on a hidden Markov model to stu...