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Velocity dispersion ($\sigma$) is a key driver for galaxy structure andevolution. We here present a comprehensive semi-empirical approach to compute$\sigma$ via detailed Jeans modelling assuming both a constant andscale-dependent mass-to-light ratio $M^*/L$. We compare with a large sample oflocal galaxies from MaNGA and find that both models can reproduce theFaber-Jackson (FJ) relation and the weak dependence of $\sigma$ onbulge-to-total ratio $B/T$ (for $B/T\gtrsim 0.25$). The dynamical-to-stellarmass ratio within $R\lesssim R_e$ can be fully accounted for by a gradient in$M^*/L$. We then build velocity dispersion evolutionary tracks$\sigma_{ap}[M^*,z]$ (within an aperture) along the main progenitor dark matterhaloes assigning stellar masses, effective radii and Sersic indices via avariety of abundance matching and empirically motivated relations. We find: 1)clear evidence for downsizing in $\sigma_{ap}[M^*,z]$ along the progenitortracks; 2) at fixed stellar mass $\sigma\propto(1+z)^{0.2-0.3}$ depending onthe presence or not of a gradient in $M^*/L$. We extract $\sigma_{ap}[M^*,z]$from the TNG50 hydrodynamic simulation and find very similar results to ourmodels with constant $M^*/L$. The increasing dark matter fraction within $R_e$tends to flatten the $\sigma_{ap}[M^*,z]$ along the progenitors at $z \gtrsim1$ in constant $M^*/L$ models, while $\sigma_{ap}[M^*,z]$ have a steeperevolution in the presence of a stellar gradient. We then show that acombination of mergers and gas accretion are likely responsible for theconstant or increasing $\sigma_{ap}[M^*,z]$ with time. Finally, our$\sigma_{ap}[M^*,z]$ are consistent with a nearly constant and steep$M_{bh}-\sigma$ relation at $z\lesssim 2$, with black hole masses derived fromthe $L_X-M^*$ relation.

The weak dependence of velocity dispersion on disk fractions, mass-to-light ratio and redshift: Implications for galaxy and black hole evolution