It has been recently shown that the halo near the Sun contains several kinematic substructures associated to past accretion events. For the more distant halo, there is evidence of large-scale density variations -- in the form of stellar clouds or overdensities. We study the link between the local halo kinematic groups and three of these stellar clouds: the Hercules-Aquila cloud, the Virgo Overdensity, and the Eridanus-Phoenix overdensity. We perform orbital integrations in a standard Milky Way potential of a local halo sample extracted from Gaia eDR3, with the goal of predicting the location of the merger debris elsewhere in the Galaxy. We specifically focus on the regions occupied by the three stellar clouds and compare their kinematic and distance distributions with those predicted from the orbits of the nearby debris. We find that the local halo substructures have families of orbits that tend to pile up in the regions where the stellar clouds have been found. The distances and velocities of the clouds member stars are in good agreement with those predicted from the orbit integrations, particularly for Gaia-Enceladus stars. This is the dominant contributor of all three overdensities, with a minor part stemming from the Helmi streams and to an even smaller extent from Sequoia. The orbital integrations predict no asymmetries in the sky distribution of halo stars, and they pinpoint where additional debris associated with the local halo substructures may be located.
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