  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.
