  We give an explicit graph formula, in terms of decorated boundary strata
classes, for the wall-crossing of universal Brill-Noether classes. More
precisely, fix $n>0$ and $d<g$ , and two stability conditions $\phi^-, \phi^+$
for degree~$d$ compactified universal (over $\overline{\mathcal{M}}_{g,n}$)
Jacobians that lie on opposite sides of a stability hyperplane. Our main result
is a formula for the difference between the Brill-Noether classes, compared via
the pullback along the (rational) identity map $\mathsf{Id} \colon
\overline{\mathcal{J}}^d_{g,n} (\phi^+) \dashrightarrow
\overline{\mathcal{J}}^d_{g,n} (\phi^-)$. The calculation involves constructing
a resolution of the identity map by means of subsequent blow-ups.
