  The similarity between Finsler and Riemann geometry is an intriguing starting
point to extend general relativity. The lack of quadratic restriction over the
line element (color) naturally generalize the Riemannian case and breaks the
local symmetries of general relativity. In addition, the Finsler manifold is
enriched with new geometric entities and all the classical identities are
suitably extended. We investigate the covariant kinematics of a medium formed
by a time-like congruence. After a brief view in the general case we impose
particular geometric restrictions to get some analytic insight. Central role to
our analysis plays the Lie derivative where even in case of irrotational
Killing vectors the bundle still deforms. We demonstrate an example of an
isotropic and exponentially expanding cross-section that finally deflates or
forms a caustic. Furthermore, using the 1+3 covariant formalism we investigate
the expansion dynamics of the congruence. For certain geometric restrictions we
retrieve the Raychaudhuri equation where a color-curvature coupling is
revealed. The condition to prevent the focusing of neighboring particles is
given and is more likely to fulfilled in highly curved regions. Then, we
introduce the Levi-Civita connection for the osculating Riemannian metric and
develop a (spatially) isotropic and homogeneous dust-like model with a
non-singular bounce.
