  We present results from a low-resolution spectroscopic survey for 21 galaxy
clusters at $0.4 < z < 0.8$ selected from the ESO Distant Cluster Survey. We
measured spectra using the low-dispersion prism in IMACS on the Magellan Baade
telescope and calculate redshifts with an accuracy of $\sigma_z = 0.007$. We
find 1763 galaxies that are brighter than $R = 22.9$ in the large-scale cluster
environs. We identify the galaxies expected to be accreted by the clusters as
they evolve to $z = 0$ using spherical infall models and find that $\sim30\%$
to $\sim70\%$ of the $z = 0$ cluster population lies outside the virial radius
at $z \sim 0.6$. For analogous clusters at $z = 0$, we calculate that the ratio
of galaxies that have fallen into the clusters since $z \sim 0.6$ to those that
were already in the core at that redshift is typically between $\sim0.3$ and
$1.5$. This wide range of ratios is due to intrinsic scatter and is not a
function of velocity dispersion, so a variety of infall histories is to be
expected for clusters with current velocity dispersions of $300
\lesssim\sigma\lesssim 1200$ km s$^{-1}$. Within the infall regions of $z \sim
0.6$ clusters, we find a larger red fraction of galaxies than in the field and
greater clustering among red galaxies than blue. We interpret these findings as
evidence of "preprocessing", where galaxies in denser local environments have
their star formation rates affected prior to their aggregation into massive
clusters, although the possibility of backsplash galaxies complicates the
interpretation.
