Analogue of continuous mapping theorem for convergence in $L^2$

Without further assumptions this is false. If $g$ is unbounded, we might not even have $g(X_n), g(X) \in L^2$.

If $g$ is bounded and continuous, then $g(X_n) \to g(X)$ in measure by the continuous mapping theorem, and also in any $L^p$ by the dominated (bounded) convergence theorem. So the statement is true in this case.

It also holds when $g$ is unbounded but Lipschitz: if $C$ is the Lipschitz constant, then $g(X_n), g(X) \in L^2$ because $|g(X_n)| \le |g(0)| + C |X_n|$ where the right side is an $L^2$ random variable. Moreover, we have $$E|g(X_n) - g(X)|^2 \le C^2 E|X_n - X|^2$$ where the latter goes to zero.

Suppose $g$ is $\alpha$-Hölder continuous, for $\alpha \geq 0$, and that $X_n \overset{L^m}{\to} X$ for some $m > 0$. If $\alpha > 0$, then $g(X_n) \overset{L^{m/\alpha}}{\to} g(X)$. If $\alpha = 0$ (i.e. bounded), and $g$ is continuous, then $g(X_n) \overset{L^m}{\to} g(X)$. The case $\alpha = 1$ corresponds to Lipschitz continuity. The case $\alpha > 0$ is very easy to prove. I'd be interested in more general results.