On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator D_B in L² depends Riesz continuously on L^∞ perturbations of local boundary conditions B. The Lipschitz bound for the map B→D_B(1+D_B²)^-1/2 depends on Lipschitz smoothness and ellipticity of B and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.