Modified dispersion relations arising in Lorentz-violating effective field theories alter the propagation of free particles and lead to observable deviations from standard relativistic kinematics. We show that, under regularity and non-degeneracy conditions, such kinematical modifications naturally induce local pseudo-Finsler geometric structures governing the free particle propagation. The pseudo-Finsler framework provides a geometric interpretation of LIV effects, encoding them in branch-dependent structures on spacetime. The construction is illustrated in the massive sectors of the Standard-Model Extension, where modified dispersion relations define local pseudo-Finsler geometries determined by the particle species acting as a spacetime probe. In this setting, we therefore construct a perturbative proof of the Kostelecký conjecture in all the physical massive subsectors of the Standard Model Extension.