Families of proper holomorphic maps

Given a smooth, open, oriented surface $X$ endowed with a family of complex structures $\{J_b{\}}_{b\in B}$ depending continuously on the parameter $b$ in a metrisable space $B$, we construct a continuous family of proper holomorphic maps $F_b:(X,J_b)\to {\mathbb{C}}^2$, $b\in B$.