Dark sector models naturally introduce elastic and dissipative self-interactions of dark matter. The heat exchange induced by the elastic scattering permits the gravothermal evolution of the halo. Through the evolution, the inner halo firstly develops a core but ultimately becomes cuspy. Based on a semi-analytic fluid model calibrated with N-body simulations, we find that a mild dissipative scattering can significantly accelerate this evolution. Using near-field dwarf galaxies that exhibit a core-like dark matter density profile, we derive novel constraints on the strength of dissipative interactions and the energy loss per collision.