with a Markov-type jump process. We namely assume that

-   a particle at position $x_i$ with probability distribution of labels $\lambda_1$ experiences a velocity field $\dot x_i=v_{\Psi^N}(x_i,\lambda_i)$ influenced by the global state of the system $\Psi^N$, the empirical measure $\frac1N\sum_{i=1}^N \delta_{(x_i,\lambda_i)}$;

-   the probability distribution of labels $\lambda_i$ evolves according to $\dot \lambda_i={\mathcal T}^*(x_i,\Psi^N)\lambda_i$, where the operator ${\mathcal T}^*$, accounting for the transitions among the labels, is the adjoint of an operator ${\mathcal T}(x,\Psi)$ defined on Lipschitz functions on $U$ and may depend on the position of the particle and on the state of the system.

In the case of two populations $U=\{F,L\}=\{\text{followers},\text{leaders}\}$, a model of this type has been recently analyzed in [@ABRS2018], as a (partial) discrete counterpart of the PDE model in [@BMPW2009]. Some simplifying assumptions had however to be added in order to perform a mean-field analysis. In particular, the velocity field

\[I11\] $$\label{I1}
v_{\Psi^N}(x_i)= \frac{1}{N}\sum_{j=1}^N K^F(x_i-x_j)\lambda_j(\{F\})+\frac{1}{N}\sum_{j=1}^N K^L(x_i-x_j)\lambda_j(\{L\})$$ did not depend on the probability on the labels $\lambda_i$ and the transition operators $$\label{I2}
\begin{split}
({\mathcal T}^*(\Psi^N)\lambda_i)(\{F\})&=-\alpha_F (\Psi^N)\lambda_i(\{F\}) + \alpha_L(\Psi^N)(1-\lambda_i(\{F\}))\,,\\
({\mathcal T}^*(\Psi^N)\lambda_i)(\{L\})&=\alpha_F (\Psi^N)\lambda_i(\{F\}) - \alpha_L(\Psi^N)(1-\lambda_i(\{F\}))
\end{split}$$

did not depend on the position $x_i$. The goal of the present paper is to get rid of these simplifying assumptions, providing 