Although powered prosthetic legs have enabled more biomimetic joint kinematics during steady-state activities like walking and stair climbing, transitions between these activities are usually handled by discretely switching controllers without considering biomimicry or the distinct role of the leading leg. This study introduces two data-driven, phase-based kinematic control approaches for seamless inter-leg transitions (i.e., initiated by either the prosthetic or intact leg) between walking and stair ascent/descent, assuming high-level knowledge of the upcoming activity. One approach employs a novel continuously-varying kinematic model that interpolates between steady-state activities as an approximate convex combination, and the other approach employs a simple switching-based model with optimized switching timing and tunable smoothing of kinematic discontinuities. Data-driven analysis indicates the continuously-varying controller remains beneficial over the switching controller for a range of classification delays. Experimental validation with a powered knee-ankle prosthesis used by two high-functioning transfemoral amputees demonstrates the continuous controller can provide more biomimetic and uninterrupted kinematic trajectories for both joints during transitions, irrespective of the initiating leg. This research underscores the potential for enabling more natural locomotion for high-functioning prosthetic leg users.