Open quantum systems are relevant for characterizing various phases of matter in a realistic setting. Yet, it remains unclear how dissipation-driven phenomena of boundary accumulation, known as the non-Hermitian skin effect and robust edge states of topological phases, synergise in a system. We study the dynamical consequences of combining the two effects in a paradigmatic Hofstadter model under dissipation. The time-dependent particle density exhibits both chiral damping, caused by the skin effect, and edge-selective extremal damping, rooted in the edge states. We further show that the time scales of the two processes decouple due to boundary-induced spectral topology, enabling both effects to appear clearly in the dynamics. We identify intermediate magnetic fields as the most favorable regime, since chiral damping is then partially recovered. Our results offer a direct dynamical probe of intertwined spectral and band topologies and are relevant for the design of robust quantum channels.