Approximation of biholomorphic maps between Runge domains by holomorphic automorphisms

We show that biholomorphic maps between certain pairs of Runge domains in the complex affine space $\mathbb C^n$, $n>1$, are limits of holomorphic automorphisms of $\mathbb C^n$. A similar result holds for volume preserving maps and also in Stein manifolds with the density property. This generalizes several results in the literature and provides a considerably simpler proof.