Suppose that 
$$\sqrt{n}(X_n - \theta)\xrightarrow{d} X,$$
according to the delta method, we have
$$\sqrt{n}(g(X_n)-g(X))\xrightarrow{d} g'(\theta)X$$
when $g$ is differentiable.

My question is, if 
$$\sqrt{n} (g_m(X)-g_m(\theta))\xrightarrow{d}g_m'(\theta)X, \quad m=1,2,\ldots $$
and $g_m(x)$ converges to $g(x)$ in some sense, are there any results indicating when we can expect$$\sqrt{n}(g(X_n)-g(\theta))\xrightarrow{d}g'(\theta)X$$