  We calculate electronic energy transport in inhomogeneous superconductors
using a fully self-consistent non-equilibrium quasiclassical Keldysh approach.
We develop a general theory and apply it a superconductor with an order
parameter that forms domain walls, of the type encountered in
Fulde-Ferrell-Larkin-Ovchinnikov state. The heat transport in the presence of a
domain wall is inherently anisotropic and non-local. Bound states in the
nonuniform region play a crucial role and control heat transport in several
ways: (i) they modify the spectrum of quasiparticle states and result in
Andreev reflection processes, and (ii) they hybridize with impurity band and
produce local transport environment with properties very different from those
in uniform superconductor. As a result of this interplay, heat transport
becomes highly sensitive to temperature, magnetic field and disorder. For
strongly scattering impurities we find that the transport across domain walls
at low temperatures is considerably more efficient than in the uniform
superconducting state.
