an element of ${\mathcal P}(U)$ is a mixed strategy) or as a space of labels (as in [@ABRS2018], where the case $U=\{\text{leader},\text{follower}\}$ was considered). A distribution of states will be described by an element $\Lambda\in{\mathcal P}(Y)$, where $Y\coloneqq {\mathbb{R}^{d}}\times{\mathcal P}(U)$. We will be concerned with the evolution of $\Lambda$, given an initial $\Lambda^0$, determined by the laws of evolution of $x$ and $\lambda$, which are going to be discussed below.

For $y=(x,\lambda)\in  Y$ and $\Psi\in{\mathcal P}_1(Y)$, we define a vector field $b_\Psi\colon  Y\to\overline Y$ through $$\label{T034}
b_\Psi(y)\coloneqq {\left(\begin{array}{c}
                    v_\Psi(y) \\
                    {\mathcal T}^*(x,\Psi)\lambda
                    \end{array}\right)}
$$ The first component of $b_\Psi$ is a velocity field in ${\mathbb{R}^{d}}$ determined by the global state of the system $\Psi$; the second component is expressed in terms of the adjoint ${\mathcal T}^*(x,\Psi)$ of an operator ${\mathcal T}(x,\Psi)$ which sees the location of the agents and the global state of the system around them. In order to state the regularity assumptions that we make on $b_\Psi$, we will discuss separately the assumptions on $v_\Psi$ and on ${\mathcal T}$.

We assume that the velocity field $v_\Psi\colon  Y\to{\mathbb{R}^{d}}$ satisfies the following conditions:

-   for every $R>0$, for every $\Psi\in{\mathcal P}(B_R^Y)$, $v_\Psi\in{\operatorname{Lip}}(B_R^{Y};{\mathbb{R}^{d}})$ uniformly with respect to $\Psi$, namely there exists 