the exquisite catalogues of proper motions and stellar positions provided by Gaia DR2, with multi-band wide-field photometry from @stetson2019. Photometry and proper motions are available for stars of M22 and $\omega\,$Centauri with radial distances smaller than $\sim$8.4 and $\sim$28.5 arcmin from the center, respectively. Most stars within $\sim$1.7 arcmin from the center of M22 and within $\sim$2.5 arcmin from the center of $\omega\,$Centauri have poor-quality Gaia DR2 proper motions because of crowding. Hence, for the stars in these central regions we used multi-band photometry and relative proper motions from [*HST*]{} images.

Gaia DR2 proper motions are affected by systematic errors that depend on the positions and the colors of the stars [e.g. @lindegren2018]. We followed the method by @vasiliev2019, which accounts for systematic errors by enlarging the uncertainties associated with proper-motion determinations. As a consequence, as discussed by @cordoni2019, the error bars provided in this work overestimate the true errors. Indeed, our main focus is the relative motion of the multiple stellar populations in $\omega\,$Centauri and M22, which share similar colors in the Gaia passbands and have, in first approximation, similar spatial distributions. Hence, the effect systematic errors on the relative motions of the distinct population may be partially cancelling 