  An efficient microcanonical dynamics has been recently introduced for Ising
spin models embedded in a generic connected graph even in the presence of
disorder i.e. with the spin couplings chosen from a random distribution. Such a
dynamics allows a coherent definition of local temperatures also when open
boundaries are coupled to thermostats, imposing an energy flow. Within this
framework, here we introduce a consistent definition for local energy currents
and we study their dependence on the disorder. In the linear response regime,
when the global gradient between thermostats is small, we also define local
conductivities following a Fourier dicretized picture. Then, we work out a
linearized "mean-field approximation", where local conductivities are supposed
to depend on local couplings and temperatures only. We compare the approximated
currents with the exact results of the nonlinear system, showing the
reliability range of the mean-field approach, which proves very good at high
temperatures and not so efficient in the critical region. In the numerical
studies we focus on the disordered cylinder but our results could be extended
to an arbitrary, disordered spin model on a generic discrete structures.
