We consider the Einstein-dust equations with positive cosmological constant $${\lambda}$$ λ on manifolds with time slices diffeomorphic to an orientable, compact 3-manifold $${S}$$ S . It is shown that the set of standard Cauchy data for the Einstein- $${\lambda}$$ λ -dust equations on $${S}$$ 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 $${{\mathcal J}^+}$$ J + that is $${C^{\infty}}$$ C ∞ if the data are of class $${C^{\infty}}$$ C ∞ 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 $${{\mathcal J}^+}$$ 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.