Let $\pi:X\to B$ be a proper submersion with spin fibers, and $E\to X$ a Hermitian vector bundle with a unitary connection $\nabla$. Then the local family index theorem for spin Dirac operator twisted by $E$ states that $$d\widetilde{\eta}=\int_{X/B}\widehat{A}(\nabla^{T^VX})\wedge ch(\nabla)-ch(\nabla^{ker(D^E)}),$$ assuming the kernel bundle exists.

I would like to know if there is an analogous statement for Chern classes? That is, replacing the Chern character by Chern classes. And also Chern classes with other Dirac operators, for example the signature operators and de Rham operator (with the corresponding conditions on the fibers).

Thank you very much in advance.