Elucidating the relationship between spin excitations and fermiology is essential for clarifying the pairing mechanism in iron-based superconductors (FeSCs). Here, we report inelastic neutron scattering results on the hole overdoped Ba_0.4K_0.6Fe₂As₂ near a Lifshitz transition, where the electron pocket at M point is nearly replaced by four hole pockets. In the normal state, the spin excitations is observed at incommensurate wave vectors with a chimney-like dispersion. By cooling down to the superconducting state, a neutron spin resonance mode emerges with a peak energy of E_r = 14-15 meV, weakly modulated along the L-direction. The incommensurability notably increases at low energies, giving rise to downward dispersions of the resonance mode. This behavior contrasts sharply with the upward dispersions of resonance observed in optimally doped Ba_0.67K_0.33Fe₂As₂ contributed by the hole to electron scattering, but resembles those in KFe₂As₂ and KCa₂Fe₄As₄F₂ where the fermiology is dominated by hole pockets. These results highlight the critical role of electronic structure modifications near the Fermi level, especially in governing interband scattering under imperfect nesting conditions, which fundamentally shape the spin dynamics of FeSCs.