Aharoni and Berger generalized Menger's Theorem to infinite graphs: For any digraph, and any subsets $A$ and $B$, there is a family $F$ of disjoint paths from $A$ to $B$ and a set separating $B$ from $A$ consisting of one vertex of each path in $F$.

But what I need is a criterion (at least, a useful sufficient condition) telling me when there is a family of disjoint paths from $A$ to $B$ covering all of $A$. (I only need it for acyclic digraphs in which every path has at most $3$ vertices.) What is known?