Moving Seshadri constants, and coverings of varieties of maximal Albanese dimension

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L o X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian 'etale covers of $X$ are arbitrarily large. As an application, given any integer $kgeq 1$, there exists an abelian 'etale cover $pcolon X' o X$ such that the adjoint system $ig|K_{X'} + p^*L ig|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.