We report on a joint theoretical and experimental study of the vibrational dynamics of the type-I clathrates A₈Sn₄₄□₂ (A = K, Rb, Cs, and □ stands for a vacancy) by high-resolution inelastic neutron scattering experiments combined with low-temperature specific heat measurements (2-300 K). Ab initio lattice dynamics calculations were performed on hypothetical vacancy-free A₈Sn₄₆ clathrates in order to determine the phonon dispersions and vibrational density of states Z(ω). The temperature dependence of the generalized vibrational density of states (GVDOS) was traced from 420 K down to 50 K, paying particular attention to the low-energy region of the GVDOS spectra. In the Cs and Rb compounds, the inelastic signal at low energies is dominated by several peaks mainly associated with the dynamics of the alkali metal atoms A in the polyhedral cages of the clathrate structure. In contrast, the low-energy spectrum of the K compound features a more pronounced contribution of the weighed modes of the framework Sn atoms. Upon cooling, the A-weighted modes soften regardless of the nature of the alkali element. The shift observed is similar for the three compounds and of small amplitude, suggestive of a dominant quasi-harmonic behavior above 50 K. The distinct dynamics of the K atoms in comparison to Cs and Rb is further demonstrated by the analyses of the low-temperature specific heat data, indicating that the low-energy Sn-weighted modes cannot be described by a Debye model with Einstein-like contributions.