Minimum equivalent digraph with respect to sources and sinks
Let's assume that $D$ contains only sources and sinks, since the input can be translated into an equivalent such input easily.

Then, note that, in any solution $D'$ for $D$, each vertex $v$ corresponds to a biclique in the underlying undirected graph $G$ of $D$ (the biclique between all sources that reach $v$ in $D'$ and all sinks that are reached from $v$ in $D'$).

I conjecture that, if $D'$ is optimal, then it contains a vertex-cut that 1:1 corresponds to an optimal biclique covering of $G$. Then, any minimum vertex-cut in $D'$ corresponds to an optimal biclique covering in $G$. However, since BICLIQUE COVER (aka BIPARTITE DIMENSION) is NP-complete, your problem is unlikely to admit a polynomial-time algorithm unless my conjecture fails.

Note that, even if my conjecture holds, technically this argument does not prove NP-hardness of your problem, since the reduction is not a Karp- but a Cook reduction.