Isogeny relations in products of families of elliptic curves

Let Eλ be the Legendre family of elliptic curves with equation Y²=X(X-1)(X-λ). Given a curve C, satisfying a condition on the degrees of some of its coordinates and parametrizing m points P₁, …, Pₘ ∈ Eλ and n points Q₁, …, Qₙ ∈ Eμ and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many points c₀ on C, such that there exists an isogeny φ: Eμ(c₀) → Eλ(c₀) and the $m+n$ points P₁(c₀), …, Pₘ(c₀), φ(Q₁(c₀)), …, φ(Qₙ(c₀)) ∈ Eλ(c₀) are linearly dependent over End(Eλ(c₀)).