We present new deep spectroscopic observations of 0.05–0.5 L* galaxies in one cluster (cl1232.5-1250) drawn from the ESO (European Southern Observatory) Distant Cluster Survey (EDisCS) sample, at z= 0.54. The new data extend the spectroscopy already available for this cluster by about 1 mag. 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 on to 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.