Context. In their evolution, star-forming galaxies are known to follow scaling relations between some fundamental physical quantities, such as the relation between mass metallicity and star formation main sequence. Aims. We study the evolution of galaxies that at a given redshift, lie simultaneously on the mass-metallicity and main-sequence relations (MZR, MSR). Methods. To this aim, we used the analytical leaky-box chemical evolution model, in which galaxy evolution is described by the infall timescale τ and the wind efficiency λ . We provide a detailed analysis of the temporal evolution of their metallicity, stellar mass, mass-weighted age, and gas fraction. Results. The evolution of the galaxies lying on the MZR and MSR at z  ∼ 0.1 suggests that the average infall timescale in two different bins of stellar masses ( M ⋆  <  10 10   M ⊙ and M ⋆  >  10 10   M ⊙ ) decreases with decreasing redshift through the addition of new galaxies with shorter timescales. This means that at each redshift, only the youngest galaxies can be assembled on the shortest timescales and still belong to the star-forming MSR. In the lowest mass bin, a decrease in median τ is accompanied by an increase in the median λ value. This implies that systems that formed at more recent times will need to eject a larger amount of mass to retain their low metallicity values. Another important result is that galactic downsizing, as traced by the age-mass relation, is naturally recovered by imposing the local MZR and MSR for star-forming galaxies. This result is retained even when a constant star formation efficiency for different galactic masses is assumed (without imposing the observed scaling relation between stellar mass and gas-depletion time-scales). Finally, we study the evolution of the hosts of C  IV -selected active galactic nuclei, which at z  ∼ 2 follow a flat MZR. When we impose that these systems lie on the MSR, we find an “inverted” MZR at lower redshifts, meaning that some additional processes must be at play in their evolution. Conclusions. In our model, galactic downsizing is a direct consequence of the MZR and MSR for star-forming galaxies. This poses a challenge for models of galaxy evolution within a cosmological framework.