Statistical inference for spatial processes from partially realized or scattered data has seen voluminous developments in diverse areas ranging from environmental sciences to business and economics. Inference on the associated rates of change has seen some recent developments. The literature has been restricted to Euclidean domains, where inference is sought on directional derivatives, rates along a chosen direction of interest, at arbitrary locations. Inference for higher order rates, particularly directional curvature has also proved useful in these settings. Modern spatial data often arise from non-Euclidean domains. This manuscript particularly considers spatial processes defined over compact Riemannian manifolds. We develop a comprehensive inferential framework for spatial rates of change for such processes over vector fields. In doing so, we formalize smoothness of process realizations and construct differential processes -- the derivative and curvature processes. We derive conditions for kernels that ensure the existence of these processes and establish validity of the joint multivariate process consisting of the ``parent'' Gaussian process (GP) over the manifold and the associated differential processes. Predictive inference on these rates is devised conditioned on the realized process over the manifold. Manifolds arise as polyhedral meshes in practice. The success of our simulation experiments for assessing derivatives for processes observed over such meshes validate our theoretical findings. By enhancing our understanding of GPs on manifolds, this manuscript unlocks a variety of potential applications in machine learning and statistics where GPs have seen wide usage. We propose a fully model-based approach to inference on the differential processes arising from a spatial process from partially observed or realized data across scattered location on a manifold.