  Employing the method of Wigner functions on curved spaces, we study classical
kinetic (Boltzmann-like) equations of distribution functions for a real scalar
field with the Lifshitz scaling. In particular, we derive the kinetic equation
for $z=2$ on general curved spaces and for $z=3$ on spatially flat spaces under
the projectability condition $N=N(t)$, where $z$ is the dynamical critical
exponent and $N$ is the lapse function. We then conjecture a form of the
kinetic equation for a real scalar field with a general dispersion relation in
general curved geometries satisfying the projectability condition, in which all
the information about the non-trivial dispersion relation is included in the
group velocity and which correctly reproduces the equations for the $z=2$ and
$z=3$ cases as well as the relativistic case. The method and equations
developed in the present paper are expected to be useful for developments of
cosmology in the context of Ho\v{r}ava-Lifshitz gravity.
