$y\coloneqq (x,\lambda)\in {\mathbb{R}^{d}}\times{\mathcal P}(U)\eqqcolon Y$. The component $x\in{\mathbb{R}^{d}}$ describes the location of an agent in space, whereas the component $\lambda\in{\mathcal P}(U)$ describes the distribution of labels of the agent. A probability distribution $\Psi\in{\mathcal P}(Y)$ denotes a distribution of agents with labels.

To define the functional setting for the dynamics, we need to consider the *free space* $$\label{P1}
{\mathcal F}(U)\coloneqq \overline{{\operatorname{span}}({\mathcal P}(U))}^{\lVert\cdot\rVert_{{\mathrm{BL}}}}\subset ({\operatorname{Lip}}(U))'\,,$$ which is also called in the literature Arens-Eells space, see [@AP; @AE1956] and [@Weaver2018 Chapter 3]. The closure in is taken with respect to the *bounded Lipschitz* norm $\lVert\cdot\rVert_{{\mathrm{BL}}}$, which is defined, for $\mu\in({\operatorname{Lip}}(U))'$, by $$\lVert \mu \rVert_{{\mathrm{BL}}}\coloneqq\sup \big\{\langle \mu,\varphi\rangle:  \varphi\in {\operatorname{Lip}}(U),  \|\varphi\|_{{\operatorname{Lip}}}{\leqslant}1\big\}.$$ With the free space ${\mathcal F}(U)$ at hand, we define $\overline Y\coloneqq {\mathbb{R}^{d}}\times{\mathcal F}(U)$ and the norm $\lVert\cdot\rVert_{\overline Y}$ by $$\label{P2}
\lVert y\rVert_{\overline Y}=\lVert(x,\lambda)\rVert_{\overline Y}\coloneqq \lvert x \rvert+\lVert \lambda \rVert_{{\mathrm{BL}}}\,;$$ and we pose $\lVert y\rVert_Y=\lVert y\rVert_{\overline Y}$.

For a given $R>0$, we denote by $B_R$ the closed ball of radius $R$ in ${\mathbb{R}^{d}}$ and by $B_R^Y$ the ball of radius $R$ in $Y$, namely $B_R^Y=\{y\in Y:\lVert y\rVert_{\overline Y}{\leqslant}R\}$, and observe that it is a compact set, since $Y$ is locally compact by our assumptions on $U$. The Banach space structure of $\overline Y\supset Y$ allows 