We present a comprehensive analysis of the stellar population properties and morphologies of red-sequence galaxies in 24 clusters and groups from z similar to 0.75 to z similar to 0.45. The dataset, consisting of 215 spectra drawn from the ESO Distant Cluster Survey, constitutes the largest spectroscopic sample at these redshifts for which such an analysis has been conducted. Analysis reveals that the evolution of the stellar population properties of red-sequence galaxies depend on their mass: while the properties of the most massive are well described by passive evolution and high-redshift formation, those of the less massive galaxies are consistent with a more extended star-formation history. We show that these scenarios reproduce the index-sigma relations and the galaxy colours. The two main results of this work are: (1) the evolution of the line-strength indices for the red-sequence galaxies can be reproduced if 40% of the galaxies with sigma < 175 km s(-1) entered the red-sequence between z = 0.75 to z = 0.45, in agreement with the fraction derived in studies of the luminosity functions; and (2) the percentage of red-sequence galaxies exhibiting early-type morphologies (E and S0) decreases by 20% from z = 0.75 to z = 0.45. This can be understood if the red-sequence becomes more populated at later times with disc galaxies whose star formation has been quenched. We conclude that the processes quenching star formation do not necessarily produce a simultaneous morphological transformation of the galaxies entering the red-sequence.