I am looking for a reference for a result of the following form:

I have a sequence of discrete probability distributions, $p_N$, where the $N$th distribution has associated state space {$k/N, 1 \leq k \leq N$}. The value $Np_N(k)$ (probability of the state $k/N$, divided by the "size" of the state) is defined by a difference equation. I want to conclude that $N p_N(Nx) \to \pi(x)$ for some density $\pi$ on $[0,1]$, described by the "limiting" ODE. (Strictly speaking, I just need to show that the set of "discrete densities" has compact closure, i.e. that all subsequences have a convergent sub-subsequence; showing that the limit of a convergent subsequence is described by the ODE is easy.)

Equivalently, I have a sequence of solutions $\tilde p_N$ (where $\tilde p_N$ corresponds to $Np_N$ in the previous paragraph) to (a sequence of) difference equations. I interpret the solutions as the values of the function $\tilde p_N$ at points $k/N, 1 \leq k \leq N$. The difference equations (and the initial conditions) "converge" to an ODE (with initial conditions), which has a unique solution. I would like to claim that the functions $\tilde p_N$ converge to this solution of the ODE. (Again, all I really need is that the functions $\tilde p_N$ converge somewhere, ideally somewhere absolutely continuous.)

Just to clarify: I'm convinced the result is true, and has been proven a dozen times elsewhere; I'm hoping someone will tell me where.