We consider the heat equation associated to Schrodinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel L-p - L-q decay estimates are established, allowing the Schrodinger operator to have a non-trivial L-2-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrodinger operators can be written as the square of Dirac-type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac-type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.