Relative monodromy of abelian logarithms for finite covers of universal families

Let's fix a complex abelian scheme endowed with a non-torsion section and with a finite surjective modular map onto a universal family of abelian varieties (with some fixed level--structure and without locally constant parts). We show that the relative monodromy group of the abelian logarithm is non-trivial and of full rank. As a consequence we deduce a new proof of Manin's kernel theorem and of the algebraic independence of the coordinates of abelian logarithms with respect to the coordinates of periods.