| In the spirit of the Mordell-Lang conjecture, we consider the intersection of a closed irreducible algebraic subvariety V of a non-isotrivial abelian scheme over a smooth irreducible base curve with the images of a finite-rank subgroup \Gamma of a fixed abelian variety A under all isogenies between A and some member of the family, where everything is defined over the field of algebraic numbers. After excluding certain degenerate cases, the André-Pink-Zannier conjecture predicts that this intersection is not Zariski dense in V. We present our results towards and sometimes beyond this conjecture under restrictions on either the dimension of V or the abelian scheme and A. |