the right-hand sides of the equations for $\sigma_i$ and $\lambda_i$ present some differences; we refer the reader to Remark \[differences\] for a more detailed comparison of the two models.

The novel contribution of this work is in specifying the tools from [@AFMS2018] to the case of our particle system . In doing this, we will pursue a more abstract point of view than the one considered in [@AFMS2018], by focusing on the structural assumptions on the velocity field $v_\Psi$ and on the operator ${\mathcal T}$ that guarantee the well-posedness of the mean-field analysis. However, we will show in Sections \[sec:LF\] and \[sec:other\] that velocities and transition rates modeled by interaction kernels as those considered in and (in this last case, with the due difference that we pointed out before) also fit in the setting that we propose in Section \[sec:abstract\_mod\]. Thus, we can retrieve an even more general version of the results of [@ABRS2018] as a by-product of our analysis (see Theorem \[T187\], where also an explicit dependence on the space variable $x$ of the transition rates is allowed).

The general assumptions on $v_\Psi$ and ${\mathcal T}$ are presented at the beginning of Section \[sec:abstract\_mod\], see (v1)-(v3) and (T0)-(T3) below. 