We report inelastic neutron scattering measurements of the phonon density of states of superconducting Sr_0.6K_0.4Fe₂As₂ (T_c=32 K) and Ca_0.6Na_0.4Fe₂As₂ (T_c=21 K). Compared with the parent compound BaFe₂As₂ doping affects mainly the lower and intermediate frequency part of the vibrations. Mass effects and lattice contraction cannot solely explain these changes. Softening of phonon modes below 10 meV has been observed in both samples on cooling from 300 to 140 K. In the Ca-doped compound the softening amounts to about 1 meV while for the Sr-doped compound the softening is about 0.5 meV. There is no appreciable change in the phonon density of states on crossing T_c.