  In this article, we study a form of free transport for the interpolated free
group factors, extending the work of Guionnet and Shlyakhtenko for the usual
free group factors. Our model for the interpolated free group factors comes
from a canonical finite von Neumann algebra $\mathcal{M}(\Gamma, \mu)$
associated to a finite, connected, weighted graph $(\Gamma,V,E, \mu)$. With
this model, we use an operator-valued version of Voiculescu's free difference
quotient to state a Schwinger-Dyson equation which is valid for the generators
of $\mathcal{M}(\Gamma, \mu)$. We construct free transport for appropriate
perturbations of this equation. Also, $\mathcal{M}(\Gamma, \mu)$ can be
constructed using the machinery of Shlyakhtenko's operator-valued semicircular
systems
