We discuss the mass-conserved reaction-diffusion system known as the wave-pinning model, which serves as a minimal framework for describing cell polarity. In this model, the interplay between reaction kinetics and slow diffusion forms a sharp interface that partitions the domain into high- and low-concentration regions. We perform a detailed asymptotic analysis and derive higher-order approximation equations governing the motion of this interface. Our results show that on a fast timescale, the interface evolves via propagating front dynamics, whereas on a slow timescale, it evolves as an area-preserving mean curvature flow. Furthermore, using the derived free boundary problem, we demonstrate that on a significantly slower timescale, an interface whose endpoints lie on the domain boundary drifts along the boundary toward regions of higher curvature.