Ubiquity of oriented rays

Call a digraph $H$ \emph{ubiquitous} if every digraph $D$ that contains $k$ vertex-disjoint copies of $H$ for every $k \in \mathbb{N}$ also contains infinitely many vertex-disjoint copies of $H$. We characterise which digraphs whose underlying undirected graph is a ray are ubiquitous.
Comment: 12 pages, 3 figures