The Sun is of course in motion with respect to other stars in our Galaxy, but it does not move quickly compared with the vast distances involved. For instance it takes about 220 million years for our Sun to orbit the Galaxy once, travelling at around 200 km/s. The stars that are closest to the Sun tend to be orbiting in more-or-less the same direction and at a similar speed (that is why they are in the vicinity of the Sun).
Thinking specifically about Polaris. There are three components of its motion with respect to the Sun - two tangential directions on the plane of the sky and a line of sight velocity.
Using the SIMBAD CDS database we see that Polaris has a line of sight velocity of 16 km/s towards the Sun and tangential motions of 28 km/s in the right ascension direction and 7 km/s in the decreasing declination direction.
This sounds a lot, but a velocity of 1 km/s means it takes about 300,000 years for the star to move 1 light year, and Polaris is about 400 light years from Earth.
So, given its net velocity, the position of Polaris with respect to us will show significant changes (of degrees) on timescales of hundreds of thousands to millions of years.
It is heading southward in the sky, but will take around 25 million years to cross the equator at the rate it is travelling now.
You say that the constellations are "constant". Yes, they are on human timescales, but the motions of the stars both radially and tangentially is routinely measured. If you waited some millions of years, the constellations would look very different.
NB: I am ignoring the precession of the Earth's rotation axis, since this is just a rotation of the coordinate system and not a change of position of the stars with respect to the Earth.
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