  In this paper, we define an energy-momentum vector at the spatial infinity of
either asymptotically flat or asymptotically hyperbolic initial data sets
carrying a non-compact boundary. Under suitable dominant energy conditions
(DECs) imposed both on the interior and along the boundary, we prove the
corresponding positive mass inequalities under the assumption that the
underlying manifold is spin. In the asymptotically flat case, we also prove a
rigidity statement when the energy-momentum vector is lightlike. Our treatment
aims to underline both the common features and the differences between the
asymptotically Euclidean and hyperbolic settings, especially regarding the
boundary DECs.
