  We present a Wigner function-based approach for the particle density
evolution in fermionic and bosonic open quantum many-body systems, including
the effects of dephasing. In particular, we focus on chains of non-interacting
particles coupled to Lindblad baths. The dissipative processes, described by
linear and quadratic jump operators, are modulated by inhomogeneous couplings.
Following a semi-classical approach, we find the differential equation
governing the Wigner function evolution, which can be solved in closed form in
some particular cases. We check the accuracy of the Wigner approach in
different scenarios (i.e. Gaussian jump rates), describing the density
evolution and the transport phenomena in terms of classical quasi-particles.
