Consider two sequences of random variables. At each point in the sequence $X_n \sim F_n$ and $Y_n \sim G_n$, and let $F_n(t)$ and $G_n(t)$ denote their respect CDFs. The distributions $(F_n, G_n)$ are not guaranteed to be continous.
Suppose that both random variables converge to the same limiting distribution. Formally, as $n \to \infty$, $X_n \to^d Z \sim H$ and $Y_n \to^d Z \sim H$, where $H(t)$ is a strictly increasing, continuous function.
Can we show that $G_n(X_n)$ converges to a uniform? If not, can we find a counter-example?