  We study thermal transport induced by soliton dynamics in a long Josephson
tunnel junction operating in the flux-flow regime. A thermal bias across the
junction is established by imposing the superconducting electrodes to reside at
different temperatures, when solitons flow along the junction. Here, we
consider the effect of both a bias current and an external magnetic field on
the thermal evolution of the device. In the flux-flow regime, a chain of
magnetically-excited solitons rapidly moves along the junction driven by the
bias current. We explore the range of bias current triggering the flux-flow
regime at fixed values of magnetic field, and the stationary temperature
distribution in this operation mode. We evidence a steady multi-peaked
temperature profile which reflects on the average soliton distribution along
the junction. Finally, we analyse also how the friction affecting the soliton
dynamics influences the thermal evolution of the system.
