On torsion sections of elliptic fibrations

Let E be an elliptic curve over the function field Q(t). Suppose that for every number field L\not=Q and every element tau\in L such that the specialization E_tau is smooth, the curve E_tau has a non-trivial torsion point over L. We show that E has a non-trivial torsion point over Q(t). This provides evidence in support of a question of Graber-Harris-Mazur-Starr on rational pseudo-sections of arithmetic surjective morphisms.