In a hierarchical Universe clusters grow via the accretion of galaxies from the field, groups and even other clusters. As this happens, galaxies can lose and/or consume their gas reservoirs via different mechanisms, eventually quenching their star formation. We explore the diverse environmental histories of galaxies through a multiwavelength study of the combined effect of ram-pressure stripping and group 'processing' in Abell 963, a massive growing cluster at z = 0.2 from the Blind Ultra Deep HI Environmental Survey (BUDHIES). We incorporate hundreds of new optical redshifts (giving a total of 566 cluster members), as well as Subaru and XMM-Newton data from LoCuSS, to identify substructures and evaluate galaxy morphology, star formation activity, and HI content (via HI deficiencies and stacking) out to 3 x R-200. We find that Abell 963 is being fed by at least seven groups, that contribute to the large number of passive galaxies outside the cluster core. More massive groups have a higher fraction of passive and HI-poor galaxies, while low-mass groups host younger (often interacting) galaxies. For cluster galaxies not associated with groups we corroborate our previous finding that HI gas (if any) is significantly stripped via ram-pressure during their first passage through the intracluster medium, and find mild evidence for a starburst associated with this event. In addition, we find an overabundance of morphologically peculiar and/or star-forming galaxies near the cluster core. We speculate that these arise from the effect of groups passing through the cluster (post-processing). Our study highlights the importance of environmental quenching and the complexity added by evolving environments.