ABSTRACT Kinematic studies of disc galaxies, using individual stars in the Milky Way or statistical studies of global disc kinematics over time, provide insight into how discs form and evolve. We use a high-resolution, cosmological zoom-simulation of a Milky Way-mass disc galaxy (h277) to tie together local disc kinematics and the evolution of the disc over time. The present-day stellar age–velocity relationship (AVR) of h277 is nearly identical to that of the analogous solar-neighbourhood measurement in the Milky Way. A crucial element of this success is the simulation’s dynamically cold multiphase ISM, which allows young stars to form with a low velocity dispersion (σbirth$\sim \!6 - 8 \ \mathrm{km\, s}^{-1}$) at late times. Older stars are born kinematically hotter (i.e. the disc settles over time in an ‘upside-down’ formation scenario), and are subsequently heated after birth. The disc also grows ‘inside-out’, and many of the older stars in the present-day solar neighbourhood are present because of radial mixing. We demonstrate that the evolution of σbirth in h277 can be explained by the same model used to describe the general decrease in velocity dispersion observed in disc galaxies from z ∼ 2–3 to the present-day, in which the disc evolves in quasi-stable equilibrium and the ISM velocity dispersion decreases over time due to a decreasing gas fraction. Thus, our results tie together local observations of the Milky Way’s AVR with observed kinematics of high z disc galaxies.