A random question.

Say we have a complex vector bundle $E\to M$ that is isomorphic to a trivial bundle, and $d$ is a trivial connection on it. Let $M\to B$ be a fiber bundle with even dimensional closed connected oriented spin fibers. If we run the process of the local family index theorem, what happens to the kernel bundle (assume it exists) and what about the connection on it? Would the $k$-fold direct sum of the kernel bundle be trivial for some large $k$?

In the literature we see some secondary index theorems but I never know what is the (primary) family index theorem for trivial (or trivialized) vector bundle.