Hexagonal Y1−xRxMnO3+δ (R: other than Y rare earth elements) oxides have been recently introduced as promising oxygen storage materials that can be utilized in the temperature swing processes for the oxygen separation and air enrichment. In the present work, the average and local structures of Tb- and Ce-substituted Y0.7Tb0.15Ce0.15MnO3+δ and Y0.6Tb0.2Ce0.2MnO3+δ materials were studied, and their oxygen storage-related properties have been evaluated. The fully oxidized samples show the presence of a significant amount of the highly oxygen-loaded the so-called Hex3 phase, attaining an average oxygen content of δ ≈ 0.41 for both compositions. Extensive studies of the temperature swing process conducted in air and N2 over the temperature range of 180–360 °C revealed large and reversible oxygen content changes taking place with only a small temperature differences and the high dependence on the oxygen partial pressure. Significant for practical performance, the highest reported for this class of compounds, oxygen storage capacity of 1900 μmol O g−1 in air was obtained for the optimized materials and swing process. In the combined temperature–oxygen partial pressure swing process, the oxygen storage capacity of 1200 μmol O g−1 was achieved.