We present a detailed analysis of the spectroscopic catalog of galaxies in
10 distant clusters from Dressier et al. We investigate the nature of the d
ifferent spectral classes defined by Dressier et al., including star-formin
g, poststarburst, and passive galaxy populations, and reproduce their basic
properties using our spectral synthesis model. We attempt to identify the
evolutionary pathways between the various spectral classes in order to sear
ch for the progenitors of the numerous poststarburst galaxies. The comparis
on of the spectra of the distant galaxy populations with samples drawn from
the local universe leads us to identify a significant population of dust-e
nshrouded starburst galaxies, showing both strong Balmer absorption and rel
atively modest [O II] emission, that we believe are the most likely progeni
tors of the poststarburst population. We present the differences between th
e field and cluster galaxies at z = 0.4-0.5. We then compare the spectral a
nd morphological properties of the distant cluster galaxies, exploring the
connection between the quenching of star formation inferred from the spectr
a and the strong evolution of the S0 population discussed by Dressier et al
. We conclude that either two different timescales and/or two different phy
sical processes are responsible for the spectral and morphological transfor
mation.