  We are concerned with a mixture of Boltzmann and McKean-Vlasov type
equations, this means (in probabilistic terms) equations with coefficients
depending on the law of the solution itself,and driven by a Poisson point
measure with the intensity depending also on the law of the solution. Both the
analytical Boltzmann equation and the probabilistic interpretation initiated by
Tanaka (1978) have intensively been discussed in the literature for specific
models related to the behavior of gas molecules. In this paper, we consider
general abstract coefficients that may include mean field effects and then we
discuss the link with specific models as well. In contrast with the usual
approach in which integral equations are used in order to state the problem, we
employ here a new formulation of the problem in terms of flows of endomorphisms
on the space of probability measure endowed with the Wasserstein distance. This
point of view already appeared in the framework of rough differential
equations. Our results concern existence and uniqueness of the solution, in the
formulation of flows, but we also prove that the "flow solution" is a solution
of the classical integral weak equation and admits a probabilistic
interpretation. Moreover, we obtain stability results and regularity with
respect to the time for such solutions. Finally we prove the convergence of
empirical measures based on particle systems to the solution of our problem,
and we obtain the rate of convergence. We discuss as examples the homogeneous
and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.
