Proper motion is the motion of nearby stars across the sky
with respect to objects that are so far away that they do
not appear to move.
Friedrich Wilhelm Bessel had taken on the work of reducing the observational data from previous astronomers, taking into account when the observations were taken, the angle through the atmosphere, various errors in the instruments, etc. Some of the early observations were 150 years earlier than some of the later observations in the data he studied. For these observations, he was able to show that some stars had apparent, or proper motion across the sky. The Fixed Stars weren't all fix after all. Thirty eight of them, at least, moved.
Some objects have very fast proper motion. For example, the planets, minor planets and comets. For these objects, it is often sufficient to take three position measurements perhaps a week apart. The Moon's proper motion can be observed in a single night. Some comets and minor planets have high speed proper motions near closest approach.
For nearby stars, one needs very good observations and spaced well apart in time. For very high precision instruments, years may suffice. However, the more time, the better.
So, how far out can an object be and still get proper motion data in a reasonable amount of time? Much of this has to do with the state of the art for measuring the positions of astronomical objects. Part of the answer lies in overcoming atmospheric disturbance. Part of the answer lies in having an instrument with a large aperture (the big end of the scope).
The Hipparcos Space Astrometry Mission was launched in August of 1989. The Hipparcos measurements of parallax is good to about 2 milliarcseconds or 0.002 arcsecond. It was a small (290 mm, 11.4 inch) telescope in space that was optimized for just this problem. It performed an end run around the atmosphere by being placed entirely above it.
Current ground based telescopes have interesting ways of coping with problems of the atmosphere and optics. Adaptive optics tracks a guide star. For some of these instruments, the guide star may be a laser. In it's simplest form, adaptive optics compensates for atmospheric turbulence in real time, using a mirror to keep the guide star in one spot.
Interferometry allows two telescopes to act as one telescope that is as large as the distance between them. The light from more than one telescope is combined to produce interference patterns, from which finer resolution data can be deduced. With interferometry, the VLT measured the size of the disk of Proxima Centauri (resolving the disk of the star!). It is no longer a point. Proxima Centauri is about 1.5 times the diameter of Jupiter. The measured angular diameter of Proxima Centauri is 1.02 milliarcseconds, that is 0.00102 arc seconds. This suggests that the precision obtained was around 0.00002 arc seconds, or about 20 micro arc seconds. Somehow, this was done from the ground, with interferometry, but without adaptive optics.
In space, the Hubble Space Telescope was used to detect the side to side wobble of a star as a planet orbits it, measuring with 0.5 milliarcseconds precision - that is, 0.0005 arc seconds. Since there is no atmosphere for the Hubble, the biggest problem is the diffraction limit for the primary mirror. Since that gives you resolving power down to about 0.05 arc seconds, you still have a factor of 100 to deal with. This factor of 100 is achieved by taking many measurements, and deducing the position of the point like star from the pattern you get. The more measurements you make, the better precision you get. There is no end to the hard work you can do. The Fine Guidance Sensors on the Hubble Space Telescope participate in the uncorrected spherical aberration of the primary mirror, but fortunately are not affected by it very much.
Interferometry can be used with radio telescopes, and the separation can be nearly the diameter of the Earth. This work achieved angular detection of 10 micro-arcseconds - that is 0.00001 arc seconds. The proper motions of the nearby galaxies M33 and M31 were recently measured. M33's speed is 190 km/s relative to the Milky Way and towards Andromeda (M31). This indicates that M33 is on a highly eccentric orbit around M31. M31 is the largest galaxy in the local group. The Milky Way (that's where we live) is the second largest. M33 is the third largest galaxy in the local group. Since M31 is heading our way, knowing it's proper motion should let us know how it will interact with the Milky Way when it gets here. That should be in around 5,000,000,000 years or so.
I'd like to watch the show when it happens.