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 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×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 as groups pass through the cluster (post-processing). Our study highlights the importance of environmental quenching and the complexity added by evolving environments.