Semi-exclusive processes like $\gamma p \to \pi^+ Y$ are closely analogous to
DIS, $ep \to eX$, in the limit where the momentum transfer $|t|$ to the pion
and the mass of the inclusive system $Y$ are large but still much smaller
than the total CM energy. We apply Bloom-Gilman duality
to this semi-exclusive process. The energy dependence of the $\gamma p \to
\pi^+ n$ cross section given by semi-local duality agrees with data for
moderate values of $|t|$, but its normalization is underestimated by about
two orders of magnitude. This indicates that rather high momentum transfers
are required for the validity of PQCD in the hard subprocess $\gamma u \to
\pi^+ d$. In the case of Compton scattering $\gamma p \to \gamma p$ the
analogous discrepancy is about one order of magnitude. In electroproduction
the virtuality of the incoming photon can be used to directly measure the
hardness of the subprocess.