Asymptotic distance between distributions in delta method

Question

Let $X_n$ be a sequence of real valued random variables such that $a_n(X_n - \mu) \xrightarrow{\text{d}} X$ for $a_n \to \infty$. Delta method states that $a_n(g(X_n) - g(\mu)) \xrightarrow{\text{d}} g'(\mu)X$, for a differentiable function $g$, in this case. What I want is an approximated distribution of $g(X_n)$ for large but finite $n$. The delta method suggests that the distributions of $g(X_n)$ and $g(\mu) + g'(\mu) X / a_n$ may be very similar for large $n$. However, I'd like to know the asymptotic behavior of the distance between the distributions. For example, $d(g(X_n), g(\mu) + g'(\mu) X / a_n) = O(f(n))$ for some definition of the distance $d(X, Y)$ between the distributions of $X$ and $Y$. Are there any useful theorems of this kind?