Pullbacks of universal Brill-Noether classes via Abel-Jacobi morphisms

Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bullet\pi_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.