The formation of buried heavily damaged and amorphous layers by a variety of swift-ion irradiations (F at 22MeV, O at 20MeV, and Mg at 28MeV) on congruent LiNbO$_3$ has been investigated. These irradiations assure that the electronic stopping power Se(z) is dominant over the nuclear stopping $S_n(z)$ and reaches a maximum value inside the crystal. The structural profile of the irradiated layers has been characterized in detail by a variety of spectroscopic techniques including dark-mode propagation, micro-Raman scattering, second-harmonic generation, and Rutherford backscattering spectroscopy∕channeling. The growth of the damage on increasing irradiation fluence presents two differentiated stages with an abrupt structural transition between them. The heavily damaged layer reached as a final stage is optically isotropic (refractive index $n = 2.10$, independent of bombarding ion) and has an amorphous structure. Moreover, it has sharp profiles and its thickness progressively increases with irradiation fluence. The dynamics under irradiation of the amorphous-crystalline boundaries has been associated with a reduction of the effective amorphization threshold due to the defects created by prior irradiation (cumulative damage). The kinetics of the two boundaries of the buried layer is quite different, suggesting that other mechanisms aside from the electronic stopping power should play a role on ion-beam damage.