We use long-slit spectroscopic observations of the sample of E+A galaxies described by Zabludoff et al . to constrain the nature of the progenitors and remnants of the E+A phase of galaxy evolution . We measure spatially-resolved kinematic properties of the young ( \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } \raise 2.0 pt% \hbox { $ \mathchar 316 $ } } 1 Gyr ) and old ( \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } \raise 2.0 pt% \hbox { $ \mathchar 318 $ } } few Gyr ) stellar populations . The young stellar populations are more centrally concentrated than the older populations , but they are not confined to the galaxy core ( radius \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } \raise 2.0 pt% \hbox { $ \mathchar 316 $ } } 1 kpc ) . The kinematics of the old stellar population place 16 of 20 of our E+As on a trend parallel to the Faber-Jackson relation that is offset by \sim 0.6 mag in R. Eighteen of 20 E+As have { v } / \sigma < 1 . As the young stars in these systems evolve , the luminosity offset will disappear and the remnants will be pressure-supported systems that lie on the Faber-Jackson relation . Although Zabludoff et al . spectroscopically selected the most extreme E+A galaxies in the local volume , the sample is kinematically diverse : velocity dispersions range from \lesssim 30 km s ^ { -1 } to \sim 200 km s ^ { -1 } over a luminosity range of M _ { R } = -19 to -22 + 5 log h . Combining these results with an estimate of the number of galaxies that experience an E+A phase , we conclude that the E+A phase of galaxy evolution is important in the development of a large fraction of spheroid-dominated galaxies over a wide range of luminosities and masses . Our kinematic observations , together with evidence that E+As have recently evolved from a vigorous star forming phase to a quiescent phase ( e.g. , Couch and Sharples , Caldwell et al . ) and that many have tidal features consistent with disk-like progenitors ( Zabludoff et al . ) , indicate that these galaxies are undergoing a transformation from gas-rich , star-forming , rotationally-supported , disk-dominated galaxies into gas-poor , quiescent , pressure-supported , spheroid-dominated galaxies .