The survival of dust grains in galaxies depends on various processes. Dust can be produced in stars, it can grow in the interstellar medium and be destroyed by astration and interstellar shocks. In this paper, we assemble a few data samples of local and distant star-forming galaxies to analyse various dust-related quantities in low- and high-redshift galaxies, and to study how the relations linking the dust mass to the stellar mass and star formation rate evolve with redshift. We interpret the available data by means of chemical evolution models for discs and proto-spheroid (PSPH) starburst galaxies. In particular, we focus on the dust-to-stellar mass (DTS) ratio, as this quantity represents a true measure of how much dust per unit stellar mass survives the various destruction processes in galaxies and is observable. The theoretical models outline the strong dependence of this quantity on the underlying star formation history. Spiral galaxies are characterized by a nearly constant DTS as a function of the stellar mass and cosmic time, whereas PSPHs present an early steep increase of the DTS, which stops at a maximal value and decreases in the latest stages. In their late starburst phase, these models show a decrease of the DTS with their mass, which allows us to explain the observed anti-correlation between the DTS and the stellar mass. The observed redshift evolution of the DTS ratio shows an increase from z ∼ 0 to z ∼ 1, followed by a roughly constant behaviour at 1 ≲ z ≲ 2.5. Our models indicate a steep decrease of the global DTS at early times, which implies an expected decrease of the DTS at larger redshift. ; Other ; PS acknowledges financial support from the European Union's Seventh Framework Programme ASTRODEEP (FP7/2007- 2013), grant agreement no. 312725.