Determining which between projected local density and distance from the cluster center plays a major role in regulating morphological fractions in clusters is a longstanding debate. Reaching a definitive answer will shed light on the main physical mechanisms at play in the most extreme environments. Here we make use of the data from the OmegaWINGS survey, currently the largest survey of clusters in the local universe extending beyond 2 virial radii from the cluster cores, to extend the previous analysis outside the virial radius. Local density and clustercentric distance seems to play different roles for galaxies of different morphology: the fraction of elliptical galaxies mainly depends on local density, suggesting that their formation was linked to the primordial densities, which now correspond to the cluster cores. Only the fraction of low-mass ellipticals shows an anticorrelation with clustercentric distance, suggesting a different origin for these objects. Excluding elliptical galaxies, the relative fraction of S0s and spirals instead depends on local density only far from the cluster cores, while within the virial radius their proportion is regulated by distance, suggesting that cluster-specific processes halt the star formation and transform Sp galaxies into S0s. This interpretation is supported by literature results on the kinematical analysis of early- and late-type galaxies, according to which fast and slow rotators have distinct dependencies on halo mass and local density.