We have used deep integral field spectroscopy obtained with the Gemini Multi-Object Spectrograph (GMOS) instrument on Gemini-North to determine the spatial distribution of the post-starburst stellar population in four luminous (̃L*) E+A galaxies at z <0.04. We find that all four galaxies have centrally concentrated gradients in the young stellar population contained within the central ̃1 kpc. This is in agreement with the Balmer line gradients found in local low-luminosity E+A galaxies. The results from higher redshift (z ̃ 0.1) samples of luminous E+A galaxies have been varied, but in general have found the post-starburst signature to be extended or a galaxy-wide phenomenon or have otherwise failed to detect gradients in the stellar populations. The ubiquity of the detection of a centrally concentrated young stellar population in local samples, and the presence of significant radial gradients in the stellar populations when the E+A galaxy core is well resolved raises the possibility that spatial resolution issues may be important in interpreting the higher redshift results. The two early-type E+A galaxies in our sample that can be robustly kinematically classified, using the λR parameter, are fast rotators. Combined with previous measurements, this brings the total number of E+A galaxies with measurements of λR to 26, with only four being classified as slow rotators. This fraction is similar to the fraction of the early-type population as a whole and argues against the need for major mergers in the production of E+A galaxies, since major mergers should result in an increased fraction of slow rotators. © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.