Given an open Riemann surface $M$, we prove that every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ ($n\geq 3$) is homotopic through nonflat conformal minimal immersions $M\to\mathbb{R}^n$ to a proper one. If $n\geq 5$, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ is homotopic to the real part of a proper holomorphic null embedding $M\to\mathbb{C}^n$. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into $\mathbb{C}^n$ directed by Oka cones in $\mathbb{C}^n$.

We prove that given a finite set $E$ in a bordered Riemann surface $\mathcal{R}$, there is a continuous map $h\colon \overline{\mathcal{R}}\setminus E\to\mathbb{C}^n$ ($n\geq 2$) such that $h|_{\mathcal{R}\setminus E} \colon \mathcal{R}\setminus E\to\mathbb{C}^n$ is a complete holomorphic immersion (embedding if $n\geq 3$) which is meromorphic on $\mathcal{R}$ and has effective poles at all points in $E$, and $h|_{b\overline{\mathcal{R}}} \colon b\overline{\mathcal{R}}\to\mathbb{C}^n$ is a topological embedding. In particular, $h(b\overline{\mathcal{R}})$ consists of the union of finitely many pairwise disjoint Jordan curves which we ensure to be of Hausdorff dimension one. We establish a more general result including uniform approximation and interpolation.