The observed properties of galaxies are strongly dependent on both their total stellar mass and their morphology. Furthermore, the environment is known to play a strong role in shaping them. The galaxy population in the local Universe that is located in virialized clusters is found to be red, poorly star-forming, and mostly composed of early morphological types. Towards a holistic understanding of the mechanisms that drive galaxy evolution, we exploit the spectrophotometric data from the WINGS and OmegaWINGS local galaxy cluster surveys, and study the role of both the local and the large-scale environments. We attempt to disentangle their effects from the intrinsic characteristics of the galaxies, in shaping the star formation activity at fixed morphological type and stellar mass. Using a sample of field galaxies from the same surveys for comparison, we analyse the effects of the environment, embodied by the local density, clustercentric distance, and close neighbours, respectively, on the star formation histories of cluster galaxies. We find that local effects have a more relevant impact on galaxy stellar properties than the large-scale environment, and that morphology needs to be taken into account to pinpoint the mechanisms that are driving the influence of clusters in galaxy evolution.