We study the intersection theory on the moduli spaces of maps of $n$-pointed curves $f:(C,s_1,\dots,s_n)\to V$ which are stable with respect to the weight data $(a_1,\dots,a_n)$, $0\le a_i\le1$. After describing the structure of these moduli spaces, we prove a formula describing the way descendant invariants change under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data $(1,\dots,1)$, and vice versa.