Fourier-Mukai transforms and normalization of nodal curves

In this article we study the relation between Arinkin's Poincar\'e sheaf on the compactified Jacobian over an integral nodal curve and the classical Poincar\'e bundle on the Jacobian of the normalization. This is applied to derive a formula for the Fourier-Mukai transform of complexes on the compactified Jacobian that come from pushforward along the normalization map. This is achieved via a third intermediate geometry: the parabolic modules introduced by Bhosle and Cook, which enode encode fiber jumps at the nodes.