We report on positron scattering processes with giant quantum systems: C60 and A@C60, where A is a rare gas atom encapsulated inside the C60 cage, employing Dirac partial wave analysis in the low-energy region. The positron-C60 interaction is simulated using a model potential approximation. Several resonances are seen in the elastic total cross sections, which cause large time delays in the associated temporal picture. The inclusion of the polarization potential alters both the resonant and non-resonant cross sections. Though it modifies the resonance details, it does not distort the origin of the resonance formation. The evolution of the resonance structures is investigated in the light of the different interparticle interactions. Absorption cross sections for rare gas endohedrals are presented in contrast to their free atomic counterparts, which elucidate structural and dynamical changes in the free atoms upon encapsulation. © 2022, The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature.