  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.
