Isogeny relations in products of families of elliptic curves

Let $E_{\lambda }$ be the Legendre family of elliptic curves with equation $Y^2=X(X-1)(X-\lambda )$. Given a curve $\mathcal{C}$, satisfying a condition on the degrees of some of its coordinates and parametrizing $m$ points $P_1,\dots ,P_m\in E_{\lambda }$ and $n$ points $Q_1,\dots ,Q_n\in E_{\mu }$ and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many points ${\mathbf{c}}_0$ on $\mathcal{C}$, such that there exists an isogeny $\phi :E_{\mu ({\mathbf{c}}_0)}\to E_{\lambda ({\mathbf{c}}_0)}$ and the $m+n$ points $P_1({\mathbf{c}}_0),\dots ,P_m({\mathbf{c}}_0),\phi (Q_1({\mathbf{c}}_0)),\dots ,\phi (Q_n({\mathbf{c}}_0))\in E_{\lambda ({\mathbf{c}}_0)}$ are linearly dependent over $\mathrm{E}\mathrm{n}\mathrm{d}(E_{\lambda ({\mathbf{c}}_0)})$.