Many complex systems show non-pairwise interactions, which can be captured by hypergraphs. In this work, we propose an edge-swapping method to sample random directed hypergraphs with fixed vertex and hyperarc degrees, which can be applied to different classes of directed hypergraphs (containing self-loops, degenerate hyperarcs and/or multi-hyperarcs). We prove that this method indeed samples uniformly from the classes with self-loops and multi-hyperarcs, and that the method may not sample uniformly from classes without self-loops, or with self-loops and degenerate hyperarcs but without multi-hyperarcs. We present a partial result on the class with self-loops, but without degenerate hyperarcs or multi-hyperarcs.