Velocity dispersion ( σ) is a key driver for galaxy structure and evolution. We here present a comprehensive semi-empirical approach to compute σvia detailed Jeans modelling assuming both a constant and scale-dependent mass-to-light ratio M ∗/ L . We compare with a large sample of local galaxies from MaNGA and find that both models can reproduce the Faber–Jackson (FJ) relation and the weak dependence of σon bulge-to-total (B/T) ratio (for B/T 0.25). The dynamical-to-stellar mass ratio within R R e can be fully accounted for by a gradient in M ∗/ L . We then build velocity dispersion evolutionary tracks σap [ M ∗, z] (within an aperture) along the main progenitor dark matter haloes assigning stellar masses, ef fecti ve radii, and S ́ersic indices via a variety of abundance matching and empirically moti v ated relations. We find: (1) clear e vidence for do wnsizing in σap [ M ∗, z] along the progenitor tracks; (2) at fixed stellar mass σ∝ (1 + z) 0.2 −0.3 depending on the presence or not of a gradient in M ∗/ L . We extract σap [ M ∗, z] from the TNG50 hydrodynamic simulation and find very similar results to our models with constant M ∗/ L . The increasing dark matter fraction within R e tends to flatten the σap [ M ∗, z] along the progenitors at z 1 in constant M ∗/ L models, while σap [ M ∗, z] have a steeper evolution in the presence of a stellar gradient. We then show that a combination of mergers and gas accretion is likely responsible for the constant or increasing σap [ M ∗, z] with time. Finally, our σap [ M ∗, z] are consistent with a nearly constant and steep M bh −σrelation at z 2, with black hole masses derived from the L X −M ∗relation.
