Summary: We consider the Einstein-dust equations with positive cosmological constant \({\lambda}\) on manifolds with time slices diffeomorphic to an orientable, compact 3-manifold \({S}\). It is shown that the set of standard Cauchy data for the Einstein-\({\lambda}\)-dust equations on \({S}\) contains an open (in terms of suitable Sobolev norms) subset of data which develop into solutions that admit at future time-like infinity a space-like conformal boundary \({{\mathcal J}^+}\) that is \({C^{\infty}}\) if the data are of class \({C^{\infty}}\) and of correspondingly lower smoothness otherwise. The class of solutions considered here comprises non-linear perturbations of FLRW solutions as very special cases. It can conveniently be characterized in terms of asymptotic end data induced on \({{\mathcal J}^+}\). These data must only satisfy a linear differential equation. If the energy density is everywhere positive they can be constructed without solving differential equations at all.