  To perform precise and accurate photometric catalogue cross-matches --
assigning counterparts between two separate datasets -- we need to describe all
possible sources of uncertainty in object position. With ever-increasing time
baselines between observations, like 2MASS in 2001 and the next generation of
surveys, such as the Vera C. Rubin Observatory's LSST, Euclid, and the Nancy
Grace Roman telescope, it is crucial that we can robustly describe and model
the effects of stellar motions on source positions in photometric catalogues.
While Gaia has revolutionised astronomy with its high-precision astrometry, it
will only provide motions for ~10% of LSST sources; additionally, LSST itself
will not be able to provide high-quality motion information for sources below
its single-visit depth, and other surveys may measure no motions at all. This
leaves large numbers of objects with potentially significant positional drifts
that may incorrectly lead matching algorithms to deem two detections too far
separated on the sky to be counterparts.
  To overcome this, in this paper we describe a model for the statistical
distribution of on-sky motions of sources of given sky coordinates and
brightness, allowing for the cross-match process to take into account this
extra potential separation between Galactic sources. We further detail how to
fold these probabilistic proper motions into Bayesian cross-matching
frameworks, such as those of Wilson & Naylor. This will vastly improve the
recovery of e.g. very red objects across optical-infrared matches, and decrease
the false match rate of photometric catalogue counterpart assignment.
