modeling the leader-follower dynamics. We apply the abstract results to this case and recover the results of [@ABRS2018], extending them to the case of $x$-dependent transition rates. Finally, Section \[discreteU\] we extend the results of Section \[sec:LF\] to any finite numbers of labels, whereas in Section \[continuousU\] we discuss some explicit examples of velocity fields $v_\Psi$ and transition operators ${\mathcal T}(x,\Psi)$ which are encompassed by our setting in the case of a continuum of labels, and we compare them with those considered in [@AFMS2018].

Preliminaries {#sec:prel}
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Basic notation
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If $(X,{\mathsf d}_X)$ is a metric space, we denote by $\mathcal{M}(X)$ the space of signed Borel measures in $X$ with finite total variation, by $\mathcal{M}_+(X)$ and ${\mathcal P}(X)$ the convex subsets of nonnegative measures and probability measures, respectively. The notation ${\mathcal P}_c(X)$ will be used for measures having compact support in $X$. For $\mu \in\mathcal{M}(X)$, $|\mu|$ denotes the total variation of $\mu$. If we denote by $C_0(X)$ the space of continuous functions vanishing at the boundary of $X$, and by $C_b(X)$ the space of bounded continuous functions, the weak$^*$ and narrow convergence in $\mathcal{M}(X)$ are defined by the convergence of the duality products $$\int_X \phi\,\mathrm{d}\mu_h \to \int_X \phi\,\mathrm{d}\mu\,,\qquad h \to 