Tensoring with a flat module $M$ will always preserve finite intersections of submodules, but not necessarily infinite intersections.
So, my question is:
Is there a known characterization of flat modules $M$ for which tensoring with $M$ preserves arbitrary intersections?
A sufficient but not necessary condition is that $M$ is a finitely generated projective module, since then tensoring with $M$ would be a continuous functor and hence preserve wide pullbacks and a fortiori, intersections of (possibly infinitely many) subobjects. But free modules show that the sufficient condition is not necessary.