Closest Star

Stars move! Which other star is closest changes over time:

A graph of the distance to seven nearby stars from 20,000 years ago to 80,000 years in the future. Each of the seven curves is a parabola. They have various widths. The curves for Alpha Centauri and Proxma Centauri nearly coincide and appear lowest (closest to Sol) on the graph, but two other curves (Ross 248 and Gliese 445) briefly dip below them. At the right edge of the graph, the Ross 128 curve croses the Centauri curves.
Image: FrancescoA, CC BY-SA 3.0, via Wikimedia Commons

Very roughly, the closest star system will be:

Until 27,000 CE Proxima Centauri
27,000 CE to 35,000 CE Alpha Centauri AB
35,000 CE to 44,000 CE Ross 248
44,000 CE to 45,000 CE Alpha Centauri AB
45,000 CE to 52,000 CE Gliese 445
52,000 CE to 80,000 CE Alpha Centauri AB
80,000 CE and for awhile Ross 128