Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps

I am interested in the following mean-field model introduced in the reference below:

There are $N$ particles. At each instant of time, a particle’s state is a particular value taken from the finite state space $Z = {0, 1, . . . , r − 1}$. The transition rate for a particle from state $i$ to state $j$ is governed by mean field dynamics: the transition rate is $λ_{i,j}(mu_N (t))$ where $mu_N (t)$ is the empirical distribution of the states of particles at time t:
begin{align}
mu_N(t)=sum_{i=1}^Ndelta_{x_i}
end{align}
The particles interact only through the dependence of their transition rates on the current empirical measure $mu_N (t)$ and therefore each particle $X_n^N(t)$ is a continuous inhomogeneous-time Markov chain with state-space $Z$.

The authors of the paper claim, without proving that, that the family $(mu_N , N geq 1)$ satisfies the weak law of large numbers in the following sense: if $mu_N(0)rightarrownu$ weakly as $Nrightarrowinfty$ for some $nuinmathcal{M}_1(Z)$, then $mu_Nrightarrowmu$ uniformly on compacts in probability, where $mu$ solves the McKean-Vlasov equation
begin{align}
dot{mu}(t) = A^* mu(t)
end{align}
with initial condition $mu(0) =nu$. Does anybody knows how to prove this claim or provide some references?

Thanks!

Reference: Vivek S. Borkar, Rajesh Sundaresan (2012) Asymptotics of the Invariant Measure in Mean Field Models with Jumps. Stochastic
Systems 2(2):322-380. https://doi.org/10.1287/12-SSY064