Let $\mathcal{X}\to S$ be a smooth family of projective varieties. Let $\beta$ be a curve class supported in some fiber and consider the relative moduli space of stable maps $\overline{M}_{g,n}(\mathcal{X},\,\beta)\to S$. Is the virtual fundamental cycle $[\overline{M}_{g,n}(\mathcal{X},\,\beta)]^{\rm vir}$ represented by a linear combination of algebraic cycles which are flat over $S$? Note that this would imply deformation-invariance of the GW invariant when these cycles have relative dimension $0$ over $S$.

If anyone has a citation, that would be very useful!