We study the Atiyah-Singer Dirac operator on smooth Riemannian Spin manifolds with smooth compact boundary. Under lower bounds on injectivity radius and bounds on the Ricci curvature and its first derivatives, we demonstrate that this operator is stable in the Riesz topology under bounded perturbations of local boundary conditions. Our work is motivated by the spectral flow and its connection to the Riesz topology. These results are obtained by obtaining similar results for a more wider class of elliptic first-order differential operators on vector bundles satisfying certain general curvature conditions. At the heart of our proofs lie methods from Calderón-Zygmund harmonic analysis coupled with the modern operator theory point of view developed in proof of the Kato square root conjecture.