There are multiple goodness of fit tests such as Kolmogorov-Smirnov, Cramer-von Mises, Anderson-Darling, and Kuiper's. Which one shall one use? There are threads (e.g. 1, 2, 3) and a nice textbook chapter (García-Portugués "Notes for Nonparametric Statistics" section 6.1) discussing that and related questions. My takeaway is that none of these tests uniformly dominates any other one in terms of power over all possible alternatives. Therefore, we cannot eliminate any of them right away. Still, if we have an idea about what the alternative is likely to be, we can choose a test that has the highest power against it.
Meanwhile, I am interested in a different aspect of comparing these test and choosing between them. Does the null distribution matter at all for the comparison? E.g. does it matter whether the null distribution is Uniform[0,1] or Normal(0,1)?
Concretely, I am interested in testing whether the probability integral transform (w.r.t. my model) of a given time series is Uniform[0,1]. (If not, that indicates my model is inadequate.) Should this knowledge influence my choice between the goodness-of-fit tests in any way?