  Recent experimental developments in multimode nonlinear photonic circuits
(MMNPC), have motivated the development of an optical thermodynamic theory that
describes the equilibrium properties of an initial beam excitation. However, a
non-equilibrium transport theory for these systems, when they are in contact
with thermal reservoirs, is still {\it terra incognita}. Here, by combining
Landauer and kinematics formalisms we develop a one-parameter scaling theory
that describes the transport in one-dimensional MMNPCs from a ballistic to a
diffusive regime. We also derive a photonic version of the Wiedemann -Franz law
that connects the thermal and power conductivities. Our work paves the way
toward a fundamental understanding of the transport properties of MMNPC and may
be useful for the design of all-optical cooling protocols.
