Although powered prosthetic legs have demonstrated energetic benefits by replicating normative joint biomechanics 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 approaches for facilitating seamless inter-leg transitions (i.e., initiated by either the prosthetic or intact leg) between walking and stair ascent/descent using a data-driven, phasebased kinematic controller. 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 an optimized switching timing and a tunable smoothing parameter to mitigate 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 transfemoral amputees demonstrates the continuous controller's superiority in providing biomimetic and uninterrupted kinematic trajectories for both joints during transitions, irrespective of the initiating leg. This research underscores the potential for improving the biomimicry of activity transitions to enable more natural locomotion for prosthetic leg users.