Moat wall (MW) pounding is prone to occurring in seismically base-isolated buildings equipped with insufficient isolation clearances due to architectural constraints, particularly during strong near-field earthquake excitations. When MW pounding occurs, the earthquake responses of the isolation layer and superstructure are significantly amplified. Hence, various hysteretic gap dampers (GDs) have been developed to exhibit additional stiffness and damping for pounding mitigation during strong earthquakes, while maintaining the isolation system's performance during small-to-moderate earthquakes. However, the effectiveness of GDs may be compromised during immediate aftershocks or future earthquakes because of their limited self-centering (SC) capabilities. For this reason, this study develops SC GDs in an adaptive isolation system, aiming to achieve optimal response control in the isolation layer and superstructure during multiple strong earthquakes through incorporating shape memory alloy (SMA) and steel GDs. Different isolation systems were designed with various GDs, i.e., SMA-, steel-, and combined-GDs, to achieve the same maximum bearing deformation under maximum considered earthquakes for comparative investigations. Seismic fragility analyses are conducted to investigate the influence of GDs' mechanical behavior on isolation effectiveness, considering mainshock-aftershock sequences. Fragility analysis results show that the designed GDs significantly reduce the probability of bearing damage compared to conventional MWs. The mechanical behavior of GDs remarkably affects the isolation performance of the adaptive isolation systems. Specifically, unlike the steel-GDs experienced unrecoverable cumulative damage, the SMA-GDs with excellent SC capability exhibit superior deformation mitigation, especially under aftershock conditions, while sacrificing higher exceedance probabilities of superstructural drifts, which is due to the inherent lower energy dissipation capability. The combined-GDs demonstrate optimal response control between the isolation layer and the superstructure.