Irregular fibrations of derived equivalent varieties

We study the behavior of irregular fibrations of a variety under derived equivalence of its bounded derived category. In particular we prove the derived invariance of the existence of an irregular fibration over a variety of general type, extending the case of irrational pencils onto curves of genus $g\geq 2$. We also prove that a derived equivalence of such fibrations induces a derived equivalence between their general fibers.