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 $L \to X$ 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 $k\geq 1$, there exists an abelian etale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ 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 $p$.