Convergence of stochastic processes via convergence of infinitesimal generators

Question

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ converge to some generator $A$ which corresponds to the process $X$. Could someone please point towards a proposition for a textbook (or if not a research paper) for me to refer to when doing this? Also, I think there are some further details here linked to the cores of the generators which I am unclear on, so it would be helpful to see a proposition/ theorem which clearly outlines the required conditions. Thanks.