There are two convergent in probability sequences of random variables $f_n \to f$ and $g_n \to g$, defined over a metric space $X$. From this post we know that under uniform continuity of $f$, the composition of sequences is convergent in probability as well, i.e. $f_n(g_n) \to f(g)$.
Is it true that
$$ \sqrt{n}(f_n(g_n) - f(g)) \to N\left(0, \mathbb{E}\left(f(g) - \mathbb{E}f(g)\right)^2\right) $$
In particular, should there be no impact from a possible dependence between $f_n$ and $g_n$ on variance?