Runge and Mergelyan theorems on families of open Riemann surfaces

Given a smooth open oriented surface $X$, endowed with a family of complex structures $\{J_b{\}}_{b\in B}$ of some Hölder class and depending continuously or smoothly on the parameter $b$ in a suitable topological space $B$, we construct continuous or smooth families $F_b:X\to Y$, $b\in B$, of $J_b$-holomorphic maps to any Oka manifold $Y$, with approximation on a suitable family of compact Runge sets in $X$. Along the way, we prove Runge and Mergelyan approximation theorems and Weierstrass interpolation theorem for functions on such families. We include applications to the construction of families of directed holomorphic immersions and conformal minimal immersions to Euclidean spaces.