Families of proper minimal surfaces

Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $\mathcal{J}=\{J_b{\}}_{b\in B}$ is a continuous family of complex structures on $X$ of local Hölder class ${\mathcal{C}}^{\alpha }$ for some $0<\alpha <1$. We construct a continuous family of $J_b$-conformal minimal immersions $u_b:X\to {\mathbb{R}}^3$, $b\in B$, properly projecting to ${\mathbb{R}}^2$ and having an arbitrary given family of flux homomorphisms ${\mathrm{F}\mathrm{l}\mathrm{u}\mathrm{x}}_{u_b}:H_1(X,\mathbb{Z})\to {\mathbb{R}}^3$. In particular, there are continuous families of proper $J_b$-holomorphic null immersions $X\to {\mathbb{C}}^3$ and of proper $J_b$-holomorphic immersions $X\to {\mathbb{C}}^2$, $b\in B$.