  We analyze diffusion processes with finite propagation speed in a
non-homogeneous medium in terms of the heterogeneous telegrapher's equation. In
the diffusion limit of infinite-velocity propagation we recover the results for
the heterogeneous diffusion process. The heterogeneous telegrapher's process
exhibits a rich variety of diffusion regimes including hyperdiffusion,
ballistic motion, superdiffusion, normal diffusion and subdiffusion, and
different crossover dynamics characteristic for complex systems in which
anomalous diffusion is observed. The anomalous diffusion exponent in the short
time limit is twice the exponent in the long time limit, in accordance to the
crossover dynamics from ballistic diffusion to normal diffusion in the standard
telegrapher's process. We also analyze the finite-velocity heterogeneous
diffusion process in presence of stochastic Poissonian resetting. We show that
the system reaches a non-equilibrium stationary state. The transition to this
non-equilibrium steady state is analysed in terms of the large deviation
function.
