as $N\to\infty$ the empirical measures $\Psi^N$ converge to a continuous path of probability measures $\Psi_t$ that solves the continuity-type equation $$\label{I5}
\partial_t\Psi_t+{\operatorname{div}}(b_{\Psi_t}\Psi_t)=0$$ in the space of probability measures on the product space $Y={\mathbb{R}^{d}}\times{\mathcal P}(U)$.

A key point of the proof is that the solutions to , which are defined via duality with test functions (Eulerian solutions, see Definition \[T038\]), can be equivalently characterized as Lagrangian solution satisfying the fixed-point equation $$\label{I6}
\Psi_t={\mathbf{Y}}_\Psi(t,0,\cdot)_\# \overline\Psi \qquad\text{for every $t{\geqslant}0$,}$$ where ${\mathbf{Y}}(t,0,\cdot)$ is the transition map (see Definition \[T142\]) associated with the ODE (with $\Psi$ in place of $\Psi^N$), $\overline\Psi$ is a given initial distribution of agents and probability on the labels, and the symbol $\#$ denotes the push-forward measure (see Definition \[pushforward\]).

In Section \[sec:LF\] we also retrieve and extend the main result of [@ABRS2018] as a special case of our analysis. We associate with a solution to the followers and leaders distributions $$\label{I7}
\mu^F_\Psi(B):=\int_{B \times {\mathcal P}(\{F, L\})} \lambda(\{F\}) \,\mathrm{d}\Psi(x, \lambda),\qquad \mu^L_\Psi(B):=\int_{B \times {\mathcal P}(\{F, L\})} \lambda(\{L\})) \,\mathrm{d}\Psi(x, \lambda),$$ for each Borel set $B \subset \mathbb{R}^d$. If the vector field $v_\Psi$ has the special structure and under suitable structural assumptions on the transition rates $\alpha_F$ and $\alpha_L$ (see below), using the definition 