Sky survey telescopes and powerful targeted telescopes play complementary roles in astronomy. In order to investigate the nature and characteristics of the motions of very faint objects, a flexibly-pointed instrument capable of high astrometric accuracy is an ideal complement to current astrometric surveys and a unique tool for precision astrophysics. Such a space-based mission will push the frontier of precision astrometry from evidence of earth-massed habitable worlds around the nearest starts, and also into distant Milky way objects up to the Local Group of galaxies. As we enter the era of the James Webb Space Telescope and the new ground-based, adaptive-optics-enabled giant telescopes, by obtaining these high precision measurements on key objects that Gaia could not reach, a mission that focuses on high precision astrometry science can consolidate our theoretical understanding of the local universe, enable extrapolation of physical processes to remote redshifts, and derive a much more consistent picture of cosmological evolution and the likely fate of our cosmos. Already several missions have been proposed to address the science case of faint objects in motion using high precision astrometry ESA missions: NEAT for M3, micro-NEAT for S1 mission, and Theia for M4 and M5. Additional new mission configurations adapted with technological innovations could be envisioned to pursue accurate measurements of these extremely small motions. The goal of this white paper is to address the fundamental science questions that are at stake when we focus on the motions of faint sky objects and to briefly review quickly instrumentation and mission profiles.Read less <