Over 30 years ago, a beautiful connection between derived categories of coherent sheaves and cohomology/motives was found for abelian schemes. More precisely, the work of Beauville and Deninger-Murre endowed the cohomology of an abelian scheme a canonical (motivic) decomposition which splits the Leray filtration; this structure, now known as the Beauville decomposition, is induced by algebraic cycles obtained from the Fourier-Mukai duality. In recent years, the study of Hitchin systems (e.g. the P=W conjecture), compactified Jacobians, and holomorphic symplectic varieties suggests that there should exist an extension of the theory of Beauville decompositions for certain abelian fibrations with singular fibers, where the Leray filtration should be replaced by the perverse filtration. I will discuss some recent progress in this direction. In particular, I will present results in both the positive and the negative directions, where Lagrangian fibrations associated with hyper-Kähler manifolds and tautological relations over the moduli of stable curves play crucial roles. Based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin.