Virtual push-forwards

Let $p:F\to G$ be a morphism of stacks of positive \emph{virtual} relative dimension $k$ and let $\gamma \in H^k(F)$. We give sufficient conditions for $p_{\ast }\gamma \cdot [F{]}^{virt}$ to be a multiple of $[G{]}^{virt}$. We apply this result to show an analogue of the conservation of number for virtually smooth families. We show implications to Gromov-Witten invariants and give a new proof of a theorem of Marian, Oprea and Pandharipande which compares the virtual classes of moduli spaces of stable maps and moduli spaces of stable quotients.