We have used the Low Resolution Imaging Spectrograph on the W. M. Keck I telescope to obtain spatially resolved spectroscopy of a small sample of six ‘post-starburst’ and three ‘dusty-starburst’ galaxies in the rich cluster CL 0016+16 at z=0.55. We use this to measure radial profiles of the Hδ and [OII]λ3727 lines which are diagnostic probes of the mechanisms that give rise to the abrupt changes in star formation rates in these galaxies. In the post-starburst sample we are unable to detect any radial gradients in the Hδ line equivalent width — although one galaxy exhibits a gradient from one side of the galaxy to the other. The absence of Hδ gradients in these galaxies is consistent with their production via interaction with the intracluster medium; however, our limited spatial sampling prevents us from drawing robust conclusions. All members of the sample have early-type morphologies, typical of post-starburst galaxies in general, but lack the high incidence of tidal tails and disturbances seen in local field samples. This argues against a merger origin and adds weight to a scenario where truncation by the intra-cluster medium is at work. The post-starburst spectral signature is consistent over the radial extent probed with no evidence of [OII]λ3727 emission and strong Hδ absorption at all radii, i.e. the post-starburst classification is not an aperture effect. In contrast the ‘dusty-starburst’ sample shows a tendency for a central concentration of [OII]λ3727 emission. This is most straightforwardly interpreted as the consequence of a central starburst. However, other possibilities exist such as a non-uniform dust distribution (which is expected in such galaxies) and/or a non-uniform starburst age distribution. The members of the sample exhibit late-type and irregular morphologies.