  Fix an abelian variety $A_0$ and a non-isotrivial abelian scheme over a
smooth irreducible curve, both defined over the algebraic numbers. Consider the
union of all images of translates of a fixed finite-rank subgroup of $A_0$,
also defined over the algebraic numbers, by abelian subvarieties of $A_0$ of
codimension at least $k$ under all isogenies between $A_0$ and some fiber of
the abelian scheme. We characterize the curves inside the abelian scheme which
are defined over the algebraic numbers, dominate the base curve and potentially
intersect this set in infinitely many points. Our proof follows the
Pila-Zannier strategy.
