Our object of study are G-connections i.e. linear partial differential equations satisfied by G-functions in several variables [A], [DGS]; typical examples are non-confluent generalized hypergeometric connections with rational parameters [GHF, Chap. 12]. Our main result is the stability of this notion under higher direct images, for any smooth morphism. As a by-product we obtain a purely p-adic proof of the open local monodromy theorem which does not use resolution of singularities.