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The accuracy of stellar distances inferred purely from parallaxes degradesrapidly with distance. Proper motion measurements, when combined with some ideaof typical velocities, provide independent information on stellar distances.Here I build a direction- and distance-dependent model of the distribution ofstellar velocities in the Galaxy, then use this together with parallaxes andproper motions to infer kinegeometric distances and transverse velocities forstars in Gaia DR3. Using noisy simulations I assess the performance of themethod and compare its accuracy to purely parallax-based (geometric) distances.Over the whole Gaia catalogue, kinegeometric distances are on average 1.25times more accurate than geometric ones. This average masks a large variationin the relative performance, however. Kinegeometric distances are considerablybetter than geometric ones beyond several kpc, for example. On average,kinegeometric distances can be measured to an accuracy of 19% and velocities(sqrt[vra^2 + vdec^2]) to 16 km/s (median absolute deviations). In Gaia DR3,kinegeometric distances are smaller than geometric ones on average for distantstars, but the pattern is more complex in the bulge and disk. With the muchmore accurate proper motions expected in Gaia DR5, a further improvement in thedistance accuracy by a factor of (only) 1.35 on average is predicted (withkinegeometric distances still 1.25 times more accurate than geometric ones).The improvement from proper motions is limited by the width of the velocityprior, in a way that the improvement from better parallaxes is not limited bythe width of the distance prior.

Estimating distances from parallaxes. VI: A method for inferring  distances and transverse velocities from parallaxes and proper motions  demonstrated on Gaia Data Release 3