The interdependence between the microstructure of sputter-deposited Ni(111)/Mo(110) superlattices and their elastic behavior is investigated as a function of the bilayer period (Lambda). Brillouin light scattering measurements show that a drastic softening of the effective shear modulus occurs with decreasing Lambda, until Lambda=2 nm where it reaches -62%. Ion irradiation is here used to trigger stress relaxation and to induce, in a controlled way, interdiffusion and structural changes allowing us thus to investigate their influence on the elastic anomaly. At a very low irradiation dose (0.1 displacements per atom), the relief of the lattice expansion and associated compressive stresses does not induce any change of the elastic response, which indicates that the elastic behavior of the as-grown multilayers is not correlated with the presence of elastic strains. Furthermore, a detailed x-ray diffraction analysis shows that the unstrained lattice parameter of Mo layers exhibit a linear dependence with the interface density, while the Ni unstrained lattice parameter remains nearly unchanged in the same Lambda range. This effect can be attributed to an interfacial mixing of a constant Ni amount (similar to1.5 monolayers) into the Mo layers, as a consequence of a dynamic segregation of Ni atoms during growth. Thus, the formation of interfacial metastable and supersaturated solid solutions, structurally and mechanically unstable, appears as the origin of the huge elastic softening observed in this system. At high ion fluences, when the mixing process becomes dominant, the present study also provides experimental data on phase transformation in ``driven'' alloys, by addressing the issue of the stability of out-of-equilibrium structures under irradiation.