We combine morphological classifications from deep Hubble Space Telescope (HST) imaging of a sample of three clusters at z = 0.31 and a further nine clusters at z = 0.37-0.56 with existing spectroscopic observations of their elliptical (E) and S0 populations, in order to study the relative spectral properties of these two galaxy types. We have also used spectroscopic and imaging data in the Coma Cluster as a present-day example of a rich cluster environment with which to compare our data at higher redshift. These data span the range in which strong evolution is claimed in the proportion of S0 galaxies within rich clusters. Techniques have recently been developed to analyze the strengths of absorption lines in the spectra of local, passive galaxies, to separate the effects of age and metallicity in the spectra and hence date the ages of the most recent substantial star formation episode in these galaxies. We show that the spectroscopic data in the distant clusters is of sufficient quality to allow us to apply these techniques and use them to determine the relative ages for the E and S0 populations in these distant clusters. We then compare the "ages" for each type in order to search for the signature of the recent formation of the bulk of the S0 population. We find no statistically significant difference between the luminosity-weighted ages of the E and S0 galaxies in these clusters. We translate this into a limit such that no more than half of the galaxies in the clusters at z = 0.31 have undergone a burst of star formation (>11% by mass) in the 1 Gyr prior to the observations. Our results, in conjunction with other work, suggest that the progenitors of the S0 galaxies in rich clusters are mostly early-type spirals that, through interactions with the cluster environment, have had their star formation truncated. This indicates a relatively unspectacular origin for the missing S0 population.