Unfavorable fluid-fluid displacement, where a low-viscosity fluid displaces a higher-viscosity fluid in permeable media, is commonly encountered in various subsurface processes. Understanding the formation and evolution of the resulting interfacial instability can have practical benefits for engineering applications. Using gradient capillary tubes as surrogate models of permeable media, we numerically investigate interfacial dynamics during gas-driven drainage. Our focus is on understanding the impact of tube geometry on interface stability. In a gradient tube, since the interface shape changes during the drainage process, we measure interfacial stability using the difference between the contact-line velocity Ucl and the meniscus tip velocity Utip. We define instability as a rapid reduction in the contact line velocity Ucl compared to the tip velocity Utip. Beyond the onset of this instability, gas penetrates into the liquid, forming a finger, and entraining a liquid film on the tube wall. The observed stability transition can be rationalized to a large extent by adaptation of an existing theory for cylindrical tubes in terms of a critical capillary number Cacrit. For an expanding tube, simulations suggest that a stability transition from an initially unstable meniscus to a final stable one, with Ucl catching up with Utip, can occur if the local capillary number is initially slightly larger than Cacrit and then drops below Cacrit. The insights gained from this study can be beneficial in estimating the mode and efficiency of subsurface fluid displacement. We numerically investigate the dynamics of a gas-liquid interface during drainage in a gradient capillary tube The observations from our numerical simulations can be rationalized by an adapted theoretical model We find a unique stabilization in drainage along expanding tubes, suppressing film entrainment even when the system is initially unstable