  In this article we study a system of $N$ particles, each of them being
defined by the couple of a position (in $\mathbb{R}^d$) and a so-called
orientation which is an element of a compact Riemannian manifold. This
orientation can be seen as a generalisation of the velocity in Vicsek-type
models such as [Degond et al. 2017; Degond, Motsch 2008]. We will assume that
the orientation of each particle follows a jump process whereas its position
evolves deterministically between two jumps. The law of the jump depends on the
position of the particle and the orientations of its neighbours. In the limit
$N\to\infty$, we first prove a propagation of chaos result which can be seen as
an adaptation of the classical result on McKean-Vlasov systems [Sznitman 1991]
to Piecewise Deterministic Markov Processes (PDMP). As in [Jourdain,
M\'el\'eard 1998], we then prove that under a proper rescaling with respect to
$N$ of the interaction radius between the agents (moderate interaction), the
law of the limiting mean-field system satisfies a BGK equation with localised
interactions which has been studied as a model of collective behaviour in
[Degond et al. 2018]. Finally, in the spatially homogeneous case, we give an
alternative approach based on martingale arguments.
