In order to understand the processes that quench star formation in cluster galaxies, we construct a library of subhalo orbits drawn from Λ cold dark matter cosmological N-body simulations of four rich clusters. We combine these orbits with models of star formation followed by environmental quenching, comparing model predictions with observed bulge and disc colours and stellar absorption line-strength indices of luminous cluster galaxies. Models in which the bulge stellar populations depend only on the galaxy subhalo mass while the disc is quenched upon infall are acceptable fits to the data. An exponential disc quenching time-scale of 3-3.5 Gyr is preferred. Quenching in lower mass groups prior to infall ('pre-processing') provides better fits, with similar quenching time-scales. Models with short (≲1 Gyr) quenching time-scales yield excessively steep cluster-centric gradients in disc colours and Balmer line indices, even if quenching is delayed for several Gyr. The data slightly prefer models where quenching occurs only for galaxies falling within ~0.5r200. These results imply that the environments of rich clusters must impact star formation rates of infalling galaxies on relatively long time-scales, indicative of gentler quenching mechanisms such as slow 'strangulation' over more rapid ram-pressure stripping. © 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.