The idea to construct a heat kernel is first construct a parametric in a small neighbourhood. Then use a bump function to extend it. And do convolution iteratively. (Reference: Laplacian on a Riemannian manifold [ROSENBERG].)

My question is will it spread all over the whole manifold? It seems that the bump function cut it off. But this should not happen for a heat kernel that it just vanish outside a neighbourhood. Because physically the heat should be able to conduct to everywhere.