Miscible CO2 soaking alternating gas (CO2-SAG) flooding is an improved version of CO2 flooding, which compensates for the insufficient interaction of CO2 and crude oil in the reservoir by adding a CO2 soaking process after the CO2 breakthrough (BT). The transmission of CO2 in the reservoir during the soaking process is controlled by the pore throat structure of the formation, which in turn affects the displacement efficiency of the subsequent secondary CO2 flooding. In this work, CO2-SAG flooding experiments at reservoir conditions (up to 70 degrees C, 18 MPa) have been carried out on four samples with very similar permeabilities but significantly different pore size distributions and pore throat structures. The results have been compared with the results of CO2 flooding on the same samples. It was found that the oil recovery factors (RFs) when using CO2-SAG flooding are higher than those when using CO2 flooding by 8-14%. In addition, we find greater improvements in the RF for rocks with greater heterogeneity of their pore throat microstructure compared with CO2 flooding. The CO2 soaking process compensates effectively for the insufficient interaction between CO2 and crude oil because of premature CO2 BT in heterogeneous cores. Moreover, rocks with a more homogeneous pore throat microstructure exhibit a higher pressure decay rate in the CO2 soaking process. The initial rapid pressure decay stage lasts for 80-135 min (in our experimental cores), accounting for over 80% of the total decay pressure. Rocks with the larger and more homogeneous pore throat microstructure exhibit smaller permeability decreases because of asphaltene precipitation after CO2-SAG flooding, possibly because the permeability of rocks with a more heterogeneous and smaller pore throat microstructure is more susceptible to damage from asphaltene precipitation. However, the overall permeability decline is 0.6-3.6% higher than that of normal CO2 flooding because of the increased time for asphaltene precipitation. Nevertheless, the corresponding permeability average decline per 1% oil RF is 0.11-0.34%, which is lower than that for CO2 flooding, making the process worthwhile. We have shown that CO2-SAG flooding has the potential to improve oil RFs with relatively less damage to cores, especially for cores with small and heterogeneous pore throat microstructures, but for which severe wettability changes due to the CO2 soaking process can become significant.