In this talk we study the behavior of special classes of fibrations onto normal projective varieties that admit a finite morphism to an abelian variety under derived equivalence of smooth projective complex varieties. Our first result is that any derived equivalence of such varieties induces a base preserving correspondence between their sets of isomorphism classes of fibrations onto smooth projective curves of genus greater or equal to two. The proof of this result involves earlier results, obtained in collaboration with M. Popa, regarding the derived invariance of the non-vanishing loci attached to the canonical bundle, and generic vanishing theory. Concerning fibrations onto higher-dimensional bases, we show how the problem of the derived invariance of fibrations is related to the conjectural derived invariance of Hodge numbers. I will report on some progress in this direction.