Existence of $K$-limits of holomorphic maps

Let $D$ be a complete hyperbolic domain in $\mathbb C^n$, $n>1$, and $N$ a compact Hermitian manifold. We prove a criterion for the existence of the $K$-limit of an arbitrary holomorphic map $f\colon D\to N$ at an arbitrary boundary point $D$ under the condition that $f$ has the corresponding radial limit at this point.