What about the NASA Pioneer 10 and 11 and Voyager 1 and 2 probes launched during the 1970s? It is to answer this question as precisely as possible that two researchers, Coryn Bailer-Jones and Davide Farnocchia, have published a study in Research Notes of the AAS.
Through complex calculations and also using data from the Gaia space telescope on the precise position of millions of stars, the two researchers have in fact traced the precise trajectories that the four spacecraft will undertake over the next thousands and millions of years and their approaches to star systems outside the solar system.
All four surveys are in the process of exiting our solar system and will approach about 60 stars over the next million years. The first real approaches (flyby) will see three of these spacecrafts (all except Pioneer 10) arrive near Proxima Centauri at distances ranging from 0.87 parsec to 1.07 parsec. This star is a red dwarf that is part of the triple star system of Alpha Centauri considered the closest star to the Sun.
Voyager 1 will flyby the nearest flyby near the star TYC 3135-52-1. The other two closest flybys will be to the stars Gaia DR2 2091429484365218432 and HD 28343. As for Voyager 2, the closest match will be the one near the star Ross 248 (0.5 parsec) while the second will be with the star Gaia DR2 4370380741264455296 (0.5 parsec).
Pioneer 10 will produce its closest encounter with the star HIP 117795 when, in about 90,000 years, it will pass to about 0.231 parsec from the star.
Pioneer 11 will make its closest flyby with the star TYC 992-192-1 (0.24 parsec) and its second closest flyby with the star Gaia DR2 454473057495679385 (0.43 parsec). In 1.2 million years, Pioneer 11 will also move to 0.8 parsec (2.6 light years) from the Delta Scuti star. This star, 202 light years away from us at the moment, just during the time of Pioneer 11’s close passage, will approach at a distance of only 11.6 light years becoming the brightest star that will appear in the Earth’s sky at that time, brighter even than Sirius today.
And as for the danger that these probes will have to face of being “captured” by any star system and therefore start an almost perennial orbit around one of these stars you can be more or less calm: researchers have calculated that such an eventuality should not happen, statistically speaking, before 1020 years (1 followed by 20 zero years or 100 billion billion years).