A number of studies based on the data collected by the Hubble Space Telescope (HST) GO-13297 program "HST Legacy Survey of Galactic Globular Clusters: Shedding UV Light on Their Populations and Formation" have investigated the photometric properties of a large sample of Galactic globular clusters and revolutionized our understanding of their stellar populations. In this paper, we expand upon previous studies by focusing our attention on the stellar clusters' internal kinematics. We computed proper motions for stars in 56 globular clusters and one open cluster by combining the GO-13297 images with archival HST data. The astrophotometric catalogs released with this paper represent the most complete and homogeneous collection of proper motions of stars in the cores of stellar clusters to date, and expand the information provided by the current (and future) Gaia data releases to much fainter stars and into the crowded central regions. We also census the general kinematic properties of stellar clusters by computing the velocity dispersion and anisotropy radial profiles of their bright members. We study the dependence on concentration and relaxation time, and derive dynamical distances. Finally, we present an in-depth kinematic analysis of the globular cluster NGC 5904.