The dependence of $v_\Psi$ on the global status of the system call for a locally Lipschitz dependence both on the status variable $(x,\lambda)$ (in a suitable topology) and on $\Psi$ with respect to the Wasserstein distance between probability measures, which is customary for the mean-field analysis of transport equation with non-local vector fields, see e.g. [@AG2008; @PR2013; @D79; @golseparticle; @villani]. Furthermore, the sublinearity condition (v3) guarantees long-time existence of the solution. Concerning the operator ${\mathcal T}$, it must comply with analogous local Lipschitz continuity conditions and sublinearity conditions, see (T1)-(T2); in addition, we require that

-   constants belong to the kernel of ${\mathcal T}(x,\Psi)$, namely ${\mathcal T}(x,\Psi)1=0$;

-   there exists $\delta>0$ such that ${\mathcal T}(x,\Psi)+\delta I{\geqslant}0$.

The two conditions above (see (T0) and (T3) below for a precise statement) ensure that the velocities $\dot\lambda_i$’s lie on the tangent plane to the probabilities on $\{F,L\}$ and that the positivity of the measures are preserved, respectively. Altogether, conditions (v1)-(v3) and (T0)-(T3) allow us to show that the particle model, which we can rewrite in the form $$\label{I4}
\dot y_i={\left(\begin{array}{c}
                    \dot x_i \\
                    \dot\lambda_i
                    \end{array}\right)}={\left(\begin{array}{c}
                    v_{\Psi^N}(x_i,\lambda_i) \\
                    {\mathcal T}^*(x_i,\Psi^N)\lambda_i
                    \end{array}\right)}\eqqcolon b_{\Psi^N}(y_i),$$ is well-posed (see Proposition \[T133\]).

We subsequently show that in the limit 