Surface terms for path integrals in field theory?

One mustn't confuse field space with physical space. The field space is some sort of manifold without boundaries (for a nonlinear sigma model), or $R^n$ for usual field theories, in either case, integration by parts works in Euclidean space, or if you add a little imaginary part to the propagators so that the action is decaying at large values of $\phi$.

The integration by parts in field space is simple--- there are no boundaries in field space, except at infinite field values, and the Euclidean or slightly off-Minkowsky action decays at infinity.

There is no relation to the integration by parts in physical space involved for instantons or other topological things.