  Let $f$ be a morphism from a klt pair $(X, \Delta)$ to an abelian variety
$A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that
$D\sim_{\mathbb Q}m(K_X+\Delta)$. We prove that the sheaf $f_*\mathcal{O}_X(D)$
becomes globally generated after pullback by an isogeny and has the Chen-Jiang
decomposition, along with some related results. These are applied to some
effective results for $\mathcal{O}_X(D)$ when $X$ is irregular.
