Birkhoff-von Neumann theorem for bistochastic digraphs
From Roger A. Horn "Topics in Matrix Analysis", section 3.2 page 164-165. 


  A matrix $Q$ is said to be doubly substochastic if its entries are nonnegative and all its row and column sums are at most one.


The doubly stochastic matrix
$\begin{bmatrix} Q &amp; I-D_r\\I-D_c &amp; Q^T\end{bmatrix}$
is a dilation of $Q$, where $D_r$ and $D_c$ are diagonal matrices with the row and column sums of $Q$ respectively. This dilation allow to reduce the generalization of the Birkhoff-von Neumann theorem for substochastic matrices to the original theorem


  Every doubly substochastic matrix is a finite convex combination of partial permutation matrices.


The bistochastic digraphs are doubly substochastic matrices, so the generalized Birkhoff-von Neumann theorem can be applied directly for them. Or one can just apply the same reduction directly. This means taking the original digraph and a copy of the digraph where all edges are reversed, and connecting the sources and sinks to their copy by suitable oriented edges of weight 1.

However, it turned out that the notion of a doubly substochastic matrix is more suitable with respect to the original motivation for this question than the notion of a bistochastic digraph. It is a good thing that the above construction directly covers that case too.