All local conservation laws of kinematic type on moving domains and moving surfaces for inviscid compressible fluid flow on curved Riemannian manifolds are derived. In particular, any such conservation laws will be found that hold only for (1) special dimensions of the manifold or the surface; (2) special conditions on the geometry of the manifold or the surface; (3) special equations of state. Importantly, the general form of these kinematic conservation laws will be allowed to depend on the intrinsic Riemannian metric, volume form, and curvature tensor of the manifold or the surface. All kinematic constants of motion that arise from the resulting kinematic conservation laws also will be determined. These results generalize earlier work on finding all kinematic local conservation laws on moving domains for inviscid compressible fluid flow in n-dimensional Euclidean space.