  We study a system consisting of $n$ particles, moving forward in jumps on the
real line. Each particle can make both independent jumps, whose sizes have some
distribution, or ``synchronization'' jumps, which allow it to join a randomly
chosen other particle if the latter happens to be ahead of it. System state is
the empirical distribution of particle locations. The mean-field asymptotic
regime, where $n\to\infty$, is considered. We prove that $v_n$, the
steady-state speed of the particle system advance, converges, as $n\to\infty$,
to a limit $v_{**}$ which can be easily found from a {\em minimum speed
selection principle.} Also, as $n\to\infty$, we prove the convergence of the
system dynamics to that of a deterministic mean-field limit (MFL). We show that
the average speed of advance of any MFL is lower bounded by $v_{**}$, and the
speed of a ``benchmark'' MFL, resulting from all particles initially
co-located, is equal to $v_{**}$.
  In the special case of exponentially distributed independent jump sizes, we
prove that a traveling wave MFL with speed $v$ exists if and only if $v\ge
v_{**}$, with $v_{**}$ having simple explicit form; we also show the existence
of traveling waves for the modified systems, with a left or right boundary
moving at a constant speed $v$. Using these traveling wave existence results,
we provide bounds on an MFL average speed of advance, depending on the right
tail exponent of its initial state. We conjecture that these results for
exponential jump sizes generalize to general jump sizes.
