Spectral convergence of Riemannian vector bundles

We consider a sequence of Hermitian vector bundles of the same rank endowed with metric connections over compact Riemannian manifolds whose heat kernels have uniform on-diagonal upper bounds, and we prove that there exists a sub-sequence of the vector bundles and a closed form on a Hilbert space to which the energy forms on the Hilbert spaces of square integrable sections of the vector bundles Mosco-converge; if, in addition, the rank is equal to one, the limit Hilbert space consists of square integrable sections of a continuous Hermitian line bundle over an open subspace in a compact metric space endowed with a Radon measure.