  We investigate of the systems of proper motions of stars in the ground-based
catalogues HSOY, UCAC5, GPS1 and PMA derived by combining with the Gaia DR1
space data. Assuming the systematic differences of stellar proper motions of
two catalogues to be caused by the mutual solid-body rotation and glide of the
coordinate systems produced by the data of the catalogues under comparison, we
analyse the components of the mutual rotation vector and displacement of the
origins of these systems. The equatorial components of the vector of mutual
rotation velocity of the compared coordinate systems, as well as velocities of
the mutual displacement of their origins, varying within the range from 0.2 to
2.9 mas/yr, were derived from a comparison of proper motions of the sources
that are common for Gaia EDR3 and the TGAS, UCAC5, HSOY, GPS1 and PMA
catalogues, respectively. The systematic errors of proper motions of stars in
the HSOY, GPS1, PMA and Gaia EDR3 catalogues in the range of faint stellar
magnitudes were estimated by analysing the formal proper motions of
extragalactic objects contained in these catalogues. The coordinate system
realised by the Gaia EDR3 data at the level of < 0.1 mas/yr is shown to have no
rotation and glide relative to the LQAC-5, ALLWISEAGN, Milliquas extragalactis
sources within the range from 15 to 21 stellar G magnitude. Among the
ground-based catalogues, the system of proper motions of the PMA stars, which
is independent of the Gaia EDR3 data, is the closest to the Gaia EDR3 system of
proper motions in G magnitude range from 15 to 21.
