It has been widely claimed that several lines of observational evidence point towards a ‘downsizing’ of the process of galaxy formation over cosmic time. This behaviour is sometimes termed ‘antihierarchical’, and contrasted with the ‘bottom-up’ (small objects form first) assembly of the dark matter structures in cold dark matter (CDM) models. In this paper, we address three different kinds of observational evidence that have been described as ‘downsizing’: the stellar mass assembly (i.e. more massive galaxies assemble at higher redshift with respect to low-mass ones), star formation rate (SFR) (i.e. the decline of the specific star formation rate is faster for more massive systems) and the ages of the stellar populations in local galaxies (i.e. more massive galaxies host older stellar populations). We compare a broad compilation of available data sets with the predictions of three different semi-analytic models of galaxy formation within the ΛCDM framework. In the data, we see only weak evidence at best of ‘downsizing’ in stellar mass and in SFR. Despite the different implementations of the physical recipes, the three models agree remarkably well in their predictions. We find that, when observational errors on stellar mass and SFR are taken into account, the models acceptably reproduce the evolution of massive galaxies (M > 1011 M⊙ in stellar mass), over the entire redshift range that we consider (0 ≲z≲ 4). However, lower mass galaxies, in the stellar mass range 109–1011 M⊙, are formed too early in the models and are too passive at late times. Thus, the models do not correctly reproduce the downsizing trend in stellar mass or the archaeological downsizing, while they qualitatively reproduce the mass-dependent evolution of the SFR. We demonstrate that these discrepancies are not solely due to a poor treatment of satellite galaxies but are mainly connected to the excessively efficient formation of central galaxies in high-redshift haloes with circular velocities ∼100–200 km s−1. We conclude that some physical processes operating on these mass scales – most probably star formation and/or supernova feedback – are not yet properly treated in these models.