ABSTRACT We combine the latest observationally motivated constraints on stellar properties in dark matter haloes, along with data-driven predictions for the atomic (H I) and molecular (H2) gas evolution in galaxies, to derive empirical relationships between the build-up of galactic components and their evolution over cosmic time. At high redshift (z ≳ 4), the frameworks imply that galaxies acquire their cold gas (both atomic and molecular) mostly by accretion, with the fraction of cold gas reaching about 20 per cent of the cosmic baryon fraction. We infer a strong dependence of the star formation rate on the H2 mass, suggesting a near-universal depletion time-scale of 0.1–1 Gyr in Milky Way-sized haloes (of masses 1012  M⊙ at z = 0). There is also evidence for a near-universality of the Kennicutt–Schmidt relation across redshifts, with very little dependence on stellar mass, if a constant conversion factor (αCO) of CO luminosity to molecular gas mass is assumed. Combining the atomic and molecular gas observations with the stellar build-up illustrates that galactic mass assembly in Milky Way-sized haloes proceeds from smooth accretion at high redshifts towards becoming merger-dominated at late times (z ≲ 0.6). Our results can be used to constrain numerical simulations of the dominant growth and accretion processes of galaxies over cosmic history.