Question

What star is closest to Earth (other than the sun) and is moving away from us (red shift)?

Answer 1

Barnad's star.

Explanation:

It is about 6 light years away and have the highest proper motion.
From wikipedia()"The star is named after the American astronomer E. E. Barnard. He was not the first to observe the star (it appeared on Harvard University plates in 1888 and 1890), but in 1916 he measured its proper motion as 10.3 arcseconds per year, which remains the largest proper motion of any star relative to the Solar System.[17]"