the sun - Are there good animations of the movement of proximate stars over long time periods?

This image should give you a good idea of the distances of stars relative to the Sun over the next 80,000 years (and the past 20,000 years).

^{Image courtesy of Wikipedia user Lambiam under the Creative Commons Attribution-Share Alike 3.0 Unported license.}

I've created a modified version below. The nearest star's track in time is in red, and the four blue boxes indicate the five transitions (the second box contains two):

Proxima Centauri remains the nearest star to us for about 25 thousand years, at which point Alpha Centauri passes it. ~7.5 thousand years after that, Ross 248 swings in closer. It then moves away after about 10 thousand years, when Alpha Centauri moves just ahead of it for a short period of time. Gliese 445 becomes the nearest star for about another ~7.5 thousand years, until it wings away and Alpha Centauri again becomes the closest star. 80 thousand years in our future, Ross 128 nudges ahead.

A few interesting things to note are that for most of this time, the nearest star is closer to the Sun than Proxima Centauri is now - indeed, there are times when quite a few stars are closer simultaneously - and that this lasts for some time as a brief spike.

However, this is only a short-term pattern. In the past, many stars have come much closer - even inside the Oort Cloud.

Also, to address the mention of Barnard's Star: It will swing close to us within ~ 10,000 years, but will just as suddenly swing away.

As an aside, a cool animation shows the movement of stars that are near the Sun throughout their past trip around the galaxy.

^{Animation courtesy of Wikipedia user Henrykus under the Creative Commons Attribution-Share Alike 3.0 Unported license. Data and original information from the European Southern Observatory.}

The above animation was the result of 1,000+ nights of observation over 15 years by astronomers working at the ESO. They observed or calculated the ages, velocities, and compositions of 14,000+ F and G stars. They then used the information to create orbital trajectories for these stars.