We study when a relatively free profinite semigroupoid, generated by a finite-vertex graph, has open multiplication. This leads to extensions of published results about finitely generated relatively free profinite semigroups. We highlight some applications that motivated our research, namely those concerning stabilizers of relatively free profinite semigroups. The proofs of these applications depend on a characterization of open multiplication in terms of what we call "fully factorizable nets" of elements in a topological semigroupoid.  This is joint work with Jorge Almeida and Herman Goulet-Ouellet.