  We introduce stochastic models for the transport of heat in systems described
by local collisional dynamics. The dynamics consists of tracer particles moving
through an array of hot scatterers describing the effect of heat baths at fixed
temperatures. Those models have the structure of Markov renewal processes. We
study their ergodic properties in details and provide a useful formula for the
cumulant generating function of the time integrated energy current. We observe
that out of thermal equilibrium, the generating function is not analytic. When
the set of temperatures of the scatterers is fixed by the condition that in
average no energy is exchanged between the scatterers and the system, different
behaviours may arise. When the tracer particles are allowed to travel freely
through the whole array of scatterers, the temperature profile is linear. If
the particles are locked in between scatterers, the temperature profile becomes
nonlinear. In both cases, the thermal conductivity is interpreted as a
frequency of collision between tracers and scatterers.
