an *appropriate functional setting* as well as a *general set of assumptions* on the velocity field $v_\Psi$ and on the operator ${\mathcal T}^*$ (see Section \[sec:abstract\_mod\]) which allow one to perform a rigorous mean-field analysis of agent-based models of the kind discussed above.

In order to develop our analysis, we will borrow some functional-analytic and some measure-theoretic tools that have been recently specified for the different, although related, context of *spatially inhomogeneous evolutionary games* in [@AFMS2018]. The analogue of is described in [@AFMS2018 Section 3.2], where the dynamics considered is $$\label{I3}
     \left\{
    \begin{aligned}
      \dot x_{i}&=a(x_{i},\sigma_{i})=\int_U e(x_{i},u)\,{\mathrm d}\sigma_{i}(u),\\
      \dot \sigma_{i}&=
      \bigg(\frac1N\sum_{j=1}^N
      \Big(\int _U J(x_{i},\cdot,x_{j},u')\,{\mathrm d}\sigma_{j}(u')-
      \int_U\int_U
      J(x_{i},w,x_{j},u')\,{\mathrm d}\sigma_{j}(u')\,{\mathrm d}\sigma_{i}(w)\Big)\bigg)
      \sigma_{i},
    \end{aligned}
    \right.$$ for $x\in{\mathbb{R}^{d}}$ tracking the position of a player and $\sigma\in{\mathcal P}(U)$ denoting their mixed strategy. In , the velocity $\dot x_i$ is determined by the $x$-dependent vector field $e(x,u)$ through an averaging process with respect to the strategies available to the player $x_i$; the evolution of the mixed strategy $\sigma_i$ is dictated by the *replicator equation* (see [@HS1998]): the term in the parenthesis in the right-hand side of the second equation above determines the performance of the strategy played by player $x_i$ with respect to all of the available 