The spectroscopic properties of galaxies in the Coma cluster are compared with those of galaxies in rich clusters at $z \sim 0.5$, to investigate the evolution of the star formation history in clusters. Luminous galaxies with $M_V \leq -20$ and post-starburst/post-starforming (k+a) spectra which constitute a significant fraction of galaxies in distant cluster samples are absent in Coma, where spectacular cases of k+a spectra are found instead at $M_V>-18.5$ and represent a significant proportion of the cluster dwarf galaxy population. A simple inspection of their positions on the sky indicates that this type of galaxy does not show a preferential location within the cluster, but the bluest and strongest-lined group of k+a's lies in projection towards the central 1.4 Mpc of Coma and have radial velocities significantly higher than the cluster mean. We find a striking correlation between the positions of these young and strong post-starburst galaxies and substructure in the hot intracluster medium (ICM) identified from {\it XMM-Newton} data, with these galaxies lying close to the edges of two infalling substructures. This result strongly suggests that the interaction with the dense ICM could be responsible for the quenching of the star formation (thus creating the k+a spectrum), and possibly, for any previous starburst. The evolution with redshift of the luminosity distribution of k+a galaxies can be explained by a ``downsizing effect'', with the maximum luminosity/mass of actively star-forming galaxies infalling onto clusters decreasing at lower redshift. We discuss the possible physical origin of this downsizing effect and the implications of our results for current scenarios of environmental effects on the star formation in galaxies.

21 pages, 7 figures, to appear in ApJ, version after referee's changes