First of all I apologize if some of the below is not clear, as I'm trying to make this as precise as possible but am not a professional mathematician...corrections or questions in the comments are much appreciated and I will correct per any guidance I receive.
Let me try to precisely define what I mean by a Markov Chain approximation to a real-valued univariate stochastic process :
A real-valued univariate stochastic process $X_t(\omega): \Omega \to \mathbb{R}$ can be approximated as a (possibly time inhomogenous) finite-state Markov Chain $M$ if there exists a sequence of finite-state-space Markov Chain models $M_n$ such that $$\forall t>0 \lim\limits_{n \to \infty} P_{X_t}[\mathrm{Range(M_{n,t})}\;\triangle \;\mathrm{Range(X_{t}})] = 0\;\mathrm{and}$$ $$\forall t>0\lim\limits_{n \to \infty}|F(M_{n,t}\leq x) - F(X_t\leq x)|=0 \;\forall x \in \mathrm{Range(X_{t}}) $$
Basically, it seems that a great many stochastic processes that model real-world dynamics can be modeled as a limit of such a sequence of finite Markov Chains.
The reason I care about this is that in applications, we often don't need absolute precision, just adequate agreement. If I can model a stochastic process (e.g., using the language of SDEs) then convert the problem to an approximation using the (much simpler and more familiar) language of Markov Chains, then it seems that a much larger range of stochastic process become amenable to analysis and numerical solutions by non-specialists in stochastic processes and practitioners.
In particular, I could use my familiarity with Markov Chains to (a) derive approximate results and (b) possibly apply limiting arguments to derive the exact results without recourse to advanced mathematics like measure theory and real analysis.
Anyway, that is my motivation for asking this, and I'd like to know the limits of this approach/idea.
Main Question: What properties of $X_t$ are necessary or sufficient (or both) for the above approximation to hold (assuming my definition of approximation is mathematically consistent (no internal contradictions or missing assumptions) and sufficient for justifying Markov Chain approximations)