Analogue of Lévy's Continuity Theorem for derivative

Question

Are there analogues of Lévy's Continuity Theorem which show that if a sequence of random variables converges in distribution, then (perhaps under extra conditions) the derivative(s) of their characteristic functions $\phi_n$ converge to the derivative(s) of the limiting random variable $\phi$? To potentially avoid conditions on certain moments existing, maybe one can establish a result of this form where $\phi_n^{(k)}(t) \to \phi^{(k)}(t)$ for all large $t$. Note $k=0$ is exactly Lévy's Continuity Theorem.