We study the two-neutrino double-β decay in 76 Ge, 116 Cd, 128 Te, 130 Te, and 150 Nd, as well as the two Gamow-Teller branches that connect the double-β decay partners with the states in the intermediate nuclei. We use a theoretical microscopic approach based on a deformed selfconsistent mean field with Skyrme interactions including pairing and spin-isospin residual forces, which are treated in a proton-neutron quasiparticle random-phase approximation. We compare our results for Gamow-Teller strength distributions with experimental information obtained from charge-exchange reactions. We also compare our results for the two-neutrino double-β decay nuclear matrix elements with those extracted from the measured half-lives. Both single-state and low-lying-state dominance hypotheses are analyzed theoretically and experimentally making use of recent data from chargeexchange reactions and β decay of the intermediate nuclei.