Using very deep, high spectral resolution data from the SAMI Integral Field Spectrograph, we study the stellar population properties of a sample of dwarf galaxies in the Fornax Cluster, down to a stellar mass of 107 M⊙, which has never been done outside the Local Group. We use full spectral fitting to obtain stellar population parameters. Adding massive galaxies from the ATLAS3D project, which we re-analysed, and the satellite galaxies of the Milky Way, we obtained a galaxy sample that covers the stellar mass range 104-1012 M⊙. Using this large range, we find that the mass-metallicity relation is not linear. We also find that the [α/Fe]-stellar mass relation of the full sample shows a U-shape, with a minimum in [α/Fe] for masses between 109 and 1010 M⊙. The relation between [α/Fe] and stellar mass can be understood in the following way: when the faintest galaxies enter the cluster environment, a rapid burst of star formation is induced, after which the gas content is blown away by various quenching mechanisms. This fast star formation causes high [α/Fe] values, like in the Galactic halo. More massive galaxies will manage to keep their gas longer and form several bursts of star formation, with lower [α/Fe] as a result. For massive galaxies, stellar populations are regulated by internal processes, leading to [α/Fe] increasing with mass. We confirm this model by showing that [α/Fe] correlates with clustercentric distance in three nearby clusters and also in the halo of the Milky Way.