Using a kinetic model of epitaxial growth, we describe how geometry controls kinetic pathways through which external deposition influences the state of a vicinal surface. The state of the surface is determined by three key, adjustable parameters: the local step angle θ, the Péclet number P, and the single-bond detachment rate . By scaling arguments in P, we find three steady-state regimes. In one regime, detailed flux balance approximately holds, so that the system is near equilibrium. In the other two regimes, geometric effects compete with deposition as the system is driven progressively out of equilibrium. Our analytical results are in excellent agreement with those of kinetic Monte Carlo simulations.