> they have roughly the same speed and direction than Sun
Are you sure about that?
https://en.wikipedia.org/wiki/Barnard%27s_Star :
"The proper motion of Barnard's Star corresponds to a relative lateral speed of 90 km/s. The 10.3 seconds of arc it travels annually amount to a quarter of a degree in a human lifetime, roughly half the angular diameter of the full Moon."
https://en.wikipedia.org/wiki/Alpha_Centauri
"All components of Alpha Centauri display significant proper motions against the background sky, similar to the first-magnitude stars Sirius and Arcturus. Over the centuries, this causes the apparent stellar positions to slowly change. Such motions define the high-proper-motion stars"
Are you sure about that?
https://en.wikipedia.org/wiki/Barnard%27s_Star : "The proper motion of Barnard's Star corresponds to a relative lateral speed of 90 km/s. The 10.3 seconds of arc it travels annually amount to a quarter of a degree in a human lifetime, roughly half the angular diameter of the full Moon."
https://en.wikipedia.org/wiki/Alpha_Centauri "All components of Alpha Centauri display significant proper motions against the background sky, similar to the first-magnitude stars Sirius and Arcturus. Over the centuries, this causes the apparent stellar positions to slowly change. Such motions define the high-proper-motion stars"