Seismic damage mechanisms of fault-crossing tunnels are complex and involve the coupled effects of fault offset and seismic ground motion. Accurately characterizing seismic demand for such tunnels remains challenging due to the limitations of conventional ground-motion intensity measures. In this study, a series of shaking-table tests is conducted to investigate the applicability of different ground-motion intensity measures (IMs) for evaluating the seismic response of fault-crossing tunnels. Both absolute-motion-based and relative-motion-based IMs are examined, and their performance is assessed in terms of response ranking consistency and correlation with tunnel lining strain demand. The experimental results show that absolute-motion-based IMs exhibit limited capability in consistently ranking tunnel response and correlating with lining strain demand under fault-crossing excitation. Relative-motion-based IMs provide improved performance; however, individual relative IMs exhibit distinct trade-offs between response ranking and correlation with response magnitude. Based on these findings, a multi relative-motion-based IM framework is proposed by combining a ratio-type IM with displacement-related IMs to jointly represent deformation characteristics and intensity levels. The proposed framework offers a more robust and physically consistent basis for characterizing fault-crossing seismic demand and provides guidance for ground-motion selection in the seismic analysis and design of fault-crossing tunnels.