We use the data from Gaia Early Data Release 3 (EDR3) to study the kinematic properties of Milky Way globular clusters. We measure the mean parallaxes and proper motions (PM) for 170 clusters, determine the PM dispersion profiles for more than 100 clusters, uncover rotation signatures in more than 20 objects, and find evidence for radial or tangential PM anisotropy in a dozen richest clusters. At the same time, we use the selection of cluster members to explore the reliability and limitations of the Gaia catalogue itself. We find that the formal uncertainties on parallax and PM are underestimated by 10-20% in dense central regions even for stars that pass numerous quality filters. We explore the spatial covariance function of systematic errors, and determine a lower limit on the uncertainty of average parallaxes and PM at the level 0.01mas and 0.025mas/yr, respectively. Finally, a comparison of mean parallaxes of clusters with distances from various literature sources suggests that the parallaxes for stars with G>13 (after applying the zero-point correction suggested by Lindegren et al. (2021A&A...649A...4L) are overestimated by ~0.01+/-0.003mas. Despite these caveats, the quality of Gaia astrometry has been significantly improved in EDR3 and provides valuable insights into the properties of star clusters.