The stars do move, but they're so far away that it would take thousands of years to be noticeable to the human eye. Astronomers can observe this movement by analyzing the shift in the wavelength of light emitted by the stars. Check the link below for a thorough.
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Correct, they move so little from our vantage point that they are referred to as "the fixed stars" in Celestial Navigation. Distances are immense, and the light has been traveling for eons. The precession of the Earth has more effect on their appearance than their actual movement.
And get this: A planet circulating around Alfa Centari (the closest star) sees the same night sky as we do! Whoa!
(All I have to do now is slip off the edge of scientific reason and assert they have the same astrology as we do and that this is the twin planet we go back and forth to when we become multi-dimensional personalities. I remember as a child thinking, "everything and everyone has suddenly changed, is it me or them, it's as if I were in an alternative dimension or on another planet."