  Using a navigation process with the datum $(F,V)$, in which $F$ is a Finsler
metric and the smooth tangent vector field $V$ satisfies $F(-V(x))>1$
everywhere, a Lorentz Finsler metric $\tilde{F}$ can be induced. Isoparametric
functions and isoparametric hypersurfaces with or without involving a smooth
measure can be defined for $\tilde{F}$. When the vector field $V$ in the
navigation datum is homothetic, we prove the local correspondences between
isoparametric functions and isoparametric hypersurfaces before and after this
navigation process. Using these correspondences, we provide some examples of
isoparametric functions and isoparametric hypersurfaces on a Funk space of
Lorentz Randers type.
