To extend the analysis to more general state dynamics that depend on the distribution of particles in a non-linear way, we can consider incorporating additional terms in the Hamiltonian or objective function that capture the measure dependence of the dynamics. This would involve modifying the HJB equation to account for the non-linear interactions between the state variables and the distribution of particles. By introducing suitable regularity conditions and assumptions on the state dynamics, we can analyze the convergence behavior of the optimal control and value functions in the presence of these non-linear dependencies. The analysis would involve studying the impact of the measure dependence on the convergence rates and the limiting behavior of the optimization process.