Descendant and Fourier-Mukai equivalences for simple flops

For a simple flop $X⇢X^{\mathrm{\prime }}$, we construct a correspondence between genus $0$ descendant Gromov-Witten theories of $X$ and $X^{\mathrm{\prime }}$. We show that the Fourier-Mukai equivalence induced by $X⇢X^{\mathrm{\prime }}$ is compatible, in a precise sense, with the descendant correspondence.