Combining HST/WFPC2 mosaics with extensive ground-based spectroscopy , we study the nature of E+A galaxies in three intermediate redshift clusters ( z = 0.33 ,~ { } 0.58 ,~ { } \&~ { } 0.83 ) . From a sample of \sim 500 confirmed cluster members , we isolate 46 E+A candidates to determine the E+A fraction and study their physical properties . Spectral types are assigned using Balmer ( H \delta , H \gamma , H \beta ) and [ OII ] \lambda 3727 equivalent widths . For all members , we have galaxy colors , luminosities , Hubble types , and quantitative structural parameters . We also include measured internal velocity dispersions for 120 cluster members , and estimate velocity dispersions for the rest of the cluster sample using the Fundamental Plane . We find E+A ’ s comprise a non-negligible component ( \sim 7 - 13 % ) of the cluster population at these redshifts , and their diverse nature indicates a heterogeneous parent population . While cluster E+A ’ s are predominantly disk-dominated systems , they span the range in Hubble type and bulge-to-total fraction to include even early-type members . Cluster E+A ’ s also cover a wide range in luminosity ( L _ { B } \sim 0.2 - 2.5 L _ { B } ^ { \ast } ) , internal velocity dispersion ( \sigma \sim 30 - 220 km s ^ { -1 } ) , and half-light radius ( r _ { 1 / 2 } \sim 0.4 - 4.3 h ^ { -1 } kpc ) . From their velocity dispersions and half-light radii , we infer that the descendants of E+A ’ s in our highest redshift cluster are massive early-type galaxies . In contrast to the wide range of luminosity and internal velocity dispersion spanned by E+A ’ s at higher redshift , only low mass E+A ’ s are found in nearby clusters , e . g . Coma . The observed decrease in the characteristic E+A mass is similar to the decrease in luminosity of rapidly star-forming field galaxies since z \sim 1 , i . e . galaxy “ down-sizing. ” In addition , we argue our statistics imply that \gtrsim 30 % of the E-S0 members have undergone an E+A phase ; the true fraction could be 100 % if the effects of E+A down-sizing , an increasing E+A fraction with redshift , and the conversion of spirals into early-types are also considered . Thus , the E+A phase may indeed be an important stage in the transformation of star-forming galaxies into early-type members .