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 similar to 30%-70% of the z = 0 cluster population lies outside the virial radius at z similar to 0.6. For analogous clusters at z = 0, we calculate that the ratio of galaxies that have fallen into the clusters since z similar to 0.6 to those that were already in the core at that redshift is typically between similar to 0.3 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 km s(-1) less than or similar to sigma less than or similar to 1200 km s(-1). Within the infall regions of z similar to 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.