We consider the Einstein constraint equations on an n-dimensional, asymptotically Euclidean manifold M with boundary. By leveraging both our own recent recent work as well as the work of some of our collaborators, we show that far-fromCMC and near-CMC solutions exist to the conformal formulation of the Einstein constraints when Robin-type marginally trapped surface boundary conditions are imposed to ensure that expansion scalars along null geodesics perpendicular to the boundary region are non-positive. Therefore, assuming a suitable form of weak cosmic censorship, the results we develop in this article provide a method to construct initial data that will evolve into a space-time containing an arbitrary number of black holes. A particularly important feature of our results are the minimal restrictions we place on the mean curvature, giving both near- and far-from-CMC results that are new.