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

On Fermionic walkers interacting with a correlated structured environment