  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.
