In Gromov–Witten theory, if the symplectic virtual fundamental classes constructed by B.Siebert satisfy functorial properties, i.e., if $f\colon X\to Y$ is an appropriate map between symplectic manifolds $X$ and $Y$, then $f_*\colon [X]^{\rm vir}=[Y]^{\rm vir}$? In his paper constructing symplectic GW invariant, I didn't see he mentions this, so does anyone knows anything about this? Thanks!

This may be not the answer you want, but in algebraic geometry there are such results, particularly in the context mentioned in Kevin Lin's comment. They usually apply to virtual classes constructed from relative, not absolute, obstruction theories. Two of them I know of are a Lemma of Kevin Costello and (particularly pertinent for Lin's comment) Cristina Manolache's applications of her own virtual pullbacks. 

