  Let $S$ and $T$ be smooth projective varieties over an algebraically closed
field. Suppose that $S$ is a surface admitting a decomposition of the diagonal.
We show that, away from the characteristic of $k$, if an algebraic
correspondence $T \to S$ acts trivially on the unramified cohomology, then it
acts trivially on any normalized, birational, and motivic functor. This
generalizes Kahn's result on the torsion order of $S$. We also exhibit an
example of $S$ over $\mathbb{C}$ for which $S \times S$ violates the integral
Hodge conjecture.
