What I have in mind are $n$ realization draws from a distribution with density $F(x)$. Then, for each $x_n$, there is a draw $y_n$ from density $G[x_n,H(F^{-1}(x_n)),y]$, with $x_n$ and $F^{-1}(x_n)$ being parameters. As $n\rightarrow \infty$, will the empirical distribution of $y_n$ converge to $G[F(x),H(1),y]$ or something?
Sorry if I wasn't able to phrase the question clearly enough, as I only have the knowledge of undergrad level probability theory.