  Almost all the media the particles move in are non-static. Depending on the
expected resolution of the studied dynamics and the amplitude of the
displacement of the media, sometimes the non-static behaviours of the media can
not be ignored. In this paper, we build the model describing L\'evy walks in
non-static media, where the physical and comoving coordinates are connected by
scale factor. We derive the equation governing the probability density function
of the position of the particles in comoving coordinate. Using the Hermite
orthogonal polynomial expansions, some statistical properties are obtained,
such as mean squared displacements (MSDs) in both coordinates and kurtosis. For
some representative non-static media and L\'{e}vy walks, the asymptotic
behaviors of MSDs in both coordinates are analyzed in detail. The stationary
distributions and mean first passage time for some cases are also discussed
through numerical simulations.
