Given a perfect obstruction theory $E^\bullet$ over a space $X$, we know that if $X$ is smooth, that the virtual fundamental class $[X, E^\bullet]$ is given by

$$[X, E^\bullet] = c_{top}\big((E^{-1})^\vee\big)$$

Suppose now that we can write $X = A \times B$ where $B$ is smooth. Let $p : X \to A$ be the projection. Is there something that we can say about $p_*[X,E^\bullet]$? In particular, if $A$ were smooth then we could compute $p_*[X, E^\bullet]$ by integration along the fibre, $B$.

If $A$ is not smooth, then can we still obtain $p_*[X, E^\bullet]$ by integrating the top chern class of a relative obstruction bundle of some kind?

What if $A$ is 0-dimensional?