  We study the dimension reduction of instantons over product manifolds with
calibrated factors. We first prove an integrability result that relates
dimension reduction with curvature conditions. Then we find a topological
criterion for bundles over product manifolds to admit instantons that satisfy
the aforementioned curvature conditions; in particular, pull-back bundles
satisfy this criterion. Consequently, we deduce explicit descriptions for the
moduli space of Hermitian Yang--Mills connections, G2-, and Spin(7)-instantons
in various contexts, and establish well-behaved compactifications for these
moduli spaces when one factor of the product manifold is a hyperk\"ahler
surface.
