Suppose we have $N$ independent 2-point Markov chains each having a rate matrix $Q = [-1,1;1,-1]$ and stationary distribution $\pi = [0.5,0.5]$. At time $t=0$, we initiate the chains so that the empirical measure $\approx \pi$. If $N$ is big, as time progresses the empirical measure will remain somewhere around $\pi$, but will exhibit some fluctuations.

For a 2-point chain, the empirical measure can be described by a single parameter, say $x$ which fixes the distribution at $[(1-x)/2, (1+x)/2]$. The initial condition says that $x(0) \approx 0$.

My question is - Is there any 'scaling law' that describes the fluctuations of $x$, as $N$ tends to $\infty$? Here by 'describes' I mean that the scaled process should converges in distribution to, say a brownian motion or solution to some appropriate stochastic differential equation with initial state $0$.

I apologise for the vague nature of the question, I'm not an expert in this area. Any pointers to references will by appreciated!