  We study the large-time behaviour of a sample $\mathcal{S}$ consisting of an
ensemble of fermionic walkers on a graph interacting with a structured infinite
reservoir of fermions $\mathcal{E}$ through an exchange of particles in
preferred states. We describe the asymptotic state of $\mathcal{S}$ in terms
the initial state of $\mathcal{E}$, with especially simple formulae in the
limit of small coupling strength. We also study the particle fluxes into the
different parts of the reservoir.
