  We discuss the geodesic motion of both massive test particles, following
timelike geodesics, and light, following null geodesics, on Finsler spacetimes
with cosmological symmetry. Using adapted coordinates on the tangent bundle of
the spacetime manifold, we derive the general form of the geodesic equation.
Further, we derive a complete set of constants of motion. As an application of
these findings, we derive the magnitude-redshift relation for light propagating
on a cosmologically symmetric Finsler background, both for a general Finsler
spacetime and for particular examples, such as spacetimes equipped with
Bogoslovsky and Randers length measures. Our results allow a confrontation of
these geometries with observations of the magnitude and redshift of supernovae.
