In order to understand the processes that quench star formation within rich clusters, we construct a library of subhalo orbits drawn from $��$CDM cosmological N-body simulations of four rich clusters. The orbits are combined with models of star formation followed by quenching in the cluster environment. These are compared with observed bulge and disc colours and stellar absorption linestrength indices of satellite galaxies. Models in which the bulge stellar populations depend only on the galaxy subhalo mass while the disc quenching depends on the cluster environment are acceptable fits to the data. An exponential disc quenching timescale of 3 - 3.5 Gyr is preferred. Models with short ($\lesssim 1$ Gyr) quenching timescales yield cluster-centric gradients in disc colours and Balmer line indices that are too steep compared to observations. We also examine models in which there is quenching in lower mass groups prior to cluster infall ("pre-processing"), finding that such models are a better fit to the data than models without pre-processing and require similar quenching times. The data slightly prefer models where quenching occurs only for galaxies falling within about 0.5 $r_{200}$. Finally, we have examined models with short quenching timescales of 1 Gyr, but a long delay time of 3 Gyr prior to quenching. All models with short quenching timescales, even such "delayed-then-rapid" quenching models, produce excessively red galaxies near the cluster core and are strongly disfavoured by the data. These results imply that the environments of rich clusters must impact star formation rates of infalling galaxies on relatively long timescales -- several times longer than a typical halo spends within the virial radius of a cluster. This scenario favours gentler quenching mechanisms such as slow "strangulation" over more rapid ram-pressure stripping.

MNRAS submitted. 18 pages, 9 figures, 1 table