In various places, an example being https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254, the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define a continuous process $Y_n(t) = \frac{1}{\sqrt{n}} X_{\lfloor nt \rfloor}$, and then show that $Y_n \Rightarrow Y$ as $n$ tends to infinity, where $Y$ is a process given by some SDE.

I'm after an expository text that explains this theory and the intuition behind this procedure. I know of two good books: Billingsley's convergence of probability measures is beginner-friendly but doesn't focus on SDEs, and Ethier and Kurtz's Markov Processes which is a great reference but perhaps not so beginner-friendly. Is there anything in between?

I asked a similar question on the stackexchange, but this received no replies. https://math.stackexchange.com/questions/3755365/expository-reference-for-scaling-limits-of-markov-processes