  We present a formalism to study many-particle quantum transport across a
lattice locally connected to two finite, non-stationary (bosonic or fermionic)
reservoirs, both of which are in a thermal state. We show that, for conserved
total particle number, a system of nonlinear quantum-classical master equations
describes the concurrent many-particle time evolution on the lattice and in the
reservoirs. The finiteness of the reservoirs makes a macroscopic current
emerge, which decreases exponentially in time, and asymptotically drives the
many-particle configuration into an equilibrium state where the particle flow
ceases. We analytically derive the time scale of this equilibration process,
and, furthermore, investigate the imprint of many-particle interferences on the
transport process.
