We investigate the stellar populations in a sample of 89 faint (M*+2 to M*+4) red galaxies in the Coma cluster, using high S/N spectroscopy from the MMT. Our sample is drawn from two 1-degree fields, one centred on the cluster core and the other located a degree to the south west of the cluster centre. For a comparison sample we use published high-S/N data for red-sequence galaxies in the Shapley Supercluster. We use state-of-the-art stellar population models to infer the SSP-equivalent age and metallicity (Fe/H) for each galaxy, as well as the abundances of the light elements Mg, Ca, C and N. The ages of the Coma dwarfs span a wide range from <2 Gyr to >10 Gyr, with a strong environmental dependence. The oldest galaxies are found only in the core, while most of the galaxies in the outer south-west field have ages ~3 Gyr. The galaxies have a metallicity range -1.0 < [Fe/H] < 0.0, and follow the same age-metallicity-mass plane as high-mass galaxies, but with increased intrinsic scatter. The Mg/Fe abundance ratios are on average slightly super-solar, and span a range -0.1 < [Mg/Fe] < +0.4. The highest Mg enhancements are found only in the cluster core, while solar ratios predominate in the outskirts. Depending on the assumed star-formation history (quenched versus burst-dominated), the number of dwarf galaxies on the red sequence in the Coma core has doubled since z~0.4-0.7. These estimates bracket the red-sequence growth timescales found by direct studies of distant clusters. In the south-west field, the red sequence was established only at z~0.1-0.2. Our observations confirm previous indications of very recently quenched star formation in this part of the cluster. Our results support the picture in which much of the cluster passive dwarf population was generated by environment-driven transformation of infalling late-type galaxies.
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