A branching random motion on a line, with abrupt changes of direction,
is studied. The branching mechanism, being independent
of random motion, and intensities of reverses are defined by a particle's
current direction. A solution of a certain hyperbolic system of coupled
non-linear equations (Kolmogorov type backward equation) has
a so-called McKean representation via such processes.
Commonly this system possesses travelling-wave solutions.
The convergence of solutions with Heaviside terminal data
to the travelling waves is discussed.
The paper realizes the McKean's program for
the Kolmogorov-Petrovskii-Piskunov equation in this case.
The Feynman-Kac formula plays a key role.