The isotopic dependence of the excitation energies of the pygmy dipole resonance (PDR) is analyzed in the framework of the self-consistent relativistic Hartree-Bogoliubov (RHB) model and the relativistic quasiparticle random-phase approximation (RQRPA). The DD-ME1 density-dependent meson-exchange interaction is used in the effective mean-field Lagrangian, and pairing correlations are described by the pairing part of the finite-range Gogny interaction D1S. Model calculations reproduce available experimental data on charge radii, the neutron skin, neutron separation energies, and excitation energies of isovector giant dipole resonances in Ni, Sn and Pb nuclei. In all three isotopic chains the one-neutron separation energies decrease with mass number much faster than the excitation energies of the PDR. As a result, already at moderate proton-neutron asymmetry the PDR peak energy is calculated above the neutron emission threshold. This result has important implications for the observation of the PDR in ([gamma],[gamma]') experiments.