The compounds of the title series have been prepared from citrate precursors under moderate pressure conditions (P = 2 GPa) and 1000 degrees C in the presence of KCIO4 as oxidizing agent. The crystal structures are cubic, space group Im (3) over bar (No. 204); the unit cell parameters linearly vary from a = 7.3272(4) angstrom (R = La) to a = 7.2409(1) angstrom (R = Lu) at room temperature. A neutron or synchrotron X-ray diffraction study of all the members of the series reveals an interesting correlation between some structural parameters and the magnetic properties. The electron injection effect upon replacement of Ca2+ with R3+ cations in the parent CaCu3Mn4O12 oxide leads to a substantial increment of the ferrimagnetic Curie temperature (T-C). An essential ingredient is supplied by the internal pressure of the R3+ cations upon a decrease in size along the rare-earth series, from La to Lu: the concomitant compression of the MnO6 octahedral units for the small rare earths provides progressively shorter Mn-O distances and improves the overlapping between Mn and O orbitals, thereby promoting superexchange and enhancing T-C by 50 K along the series. This interaction is also reinforced by a ferromagnetic component that depends on the local distortion of the MnO6 octahedra, which also increases along the series, constituting an additional factor, via intersite virtual charge transfer t-e orbital hybridization, for the observed increment of T-C.