  We present new deep spectroscopic observations of 0.05--0.5 L_* galaxies in
one cluster (cl1232.5-1250) drawn from the ESO Distant Cluster Survey (EDisCS)
sample, at z = 0.54. The new data extend the spectroscopy already available for
this cluster by about 1 magnitude. The cluster has a large fraction of passive
galaxies and exhibits a well defined and relatively tight colour-magnitude
relation. Among spectroscopic members, only six galaxies are classified as
`post-starburst' (k+a). For another EDisCS cluster at similar redshift and with
as deep spectroscopy, no member is found to have a k+a spectrum. The low
measured numbers of post-starburst systems appear to be inadequate to explain
the observed increase of faint red galaxies at lower redshift, even when
accounting for the infall of new galaxies onto the cluster. Post-starburst
galaxies represent a possible channel to move galaxies from the blue
star-forming cloud to the red-sequence, but the available data suggest this is
not the dominant channel in galaxy clusters. If the galaxies at the faint end
of the red-sequence in nearby clusters originate from the blue galaxies
observed in distant galaxy clusters, the transformation must have occurred
primarily through physical processes that do not lead to a post-starburst
phase. In addition, our data exclude a large population of k+a galaxies at
faint magnitudes, as found in the Coma cluster.
