Interactions between fluid flows and evolving boundaries are at the heart of some of the most challenging problems in applied mathematics, geophysics, climate physics, and engineering. In this talk, I will explore the interplay between shear- and buoyancy-driven flows and a phase-changing boundary using linear stability analysis and direct numerical simulations. Specifically, I will consider Rayleigh-Benard-Couette and Rayleigh-Benard-Poiseuille flows of a pure liquid phase under its solid phase - which can either melt or freeze - to understand the impact of these interactions on the evolution of the solid phase and the heat transport in the liquid phase.