  We study a model for flocking given by a $n$-particle system under which each
particle jumps forward by a random amount, independently sampled from a given
distribution $\theta$, with rate given by a non-increasing function $w$ of its
signed distance from the system center of mass. This model was introduced in
Bal\'azs et. al. (2014) and some of its properties were studied for the case
when $w$ is bounded. In the current work we are interested in the setting where
$w$ is unbounded, and this feature results in a stochastic dynamical system for
interacting particles with fast and large jumps for which little is available
in the literature. We characterize the large $n$ limit (the so-called `fluid
limit') of the empirical measure process associated with the system and prove a
propagation of chaos result. Next, for the centered $n$-particle system, by
constructing suitable Lyapunov functions, we establish existence and uniqueness
of stationary distributions and study their tail properties. In the special
case where $w$ is an exponential function and $\theta$ is an exponential
distribution, by establishing that all stationary solutions of the
McKean-Vlasov equation must be the unique fixed point of the equation, we prove
a propagation of chaos result at $t=\infty$ and establish convergence of the
particle system, starting from stationarity, in the large $n$ limit, to a
traveling wave solution of the McKean-Vlasov equation. The proof of this result
may be of interest for other interacting particle systems where convexity
properties or functional inequalities generally used for establishing such a
result are not available. Our work answers several open problems posed in
Bal\'azs et. al.
