I came across the paper of van der Vaart and Wellner (link). The problem I am facing is to derive the convergence of the probability integral transform to another distribution. Suppose we have two distributions of a random variable $X$, given by $F_X$ and $F_{X'}$, with regular properties, well-defined with existing PDFs. The latter can be thought of as, for instance, a distribution of $X$ conditional on some event, or simply a functional transformation of $F_X$.
Take a functional form represented by $F_X^{-1} (F_{X'})$, where the inverse is the quantile function of $X$, maping from $[0,1]$ onto a bounded set $[a,b]$. Relating to the proposed paper, can how would you derive the convergence of
$$ \sqrt{n}P\left(F_X^{-1} (\hat{F}_{X'}) - F_X^{-1} (F_{X'})\right), $$
where $\hat{F}_{X'}$ is the empirical measure of $F_{X'}$?