Flat-Earthers often claim that the fact Polaris not appearing to move is ‘evidence’ that the Earth is flat and stationary. If the Earth is a rotating sphere, then Polaris —as they say— should appear to be in motion.
In reality, Polaris is indeed moving across the sky. Though it is not something we can observe in a single night, or even in our entire lifetime.
Because of Earth’s rotation, stars appear to rotate in almost 24 hours cycle, similar to the Sun, though not exactly the same. These stars appear to rotate around the north and south celestial pole.
Polaris is close to the north celestial pole, and thus appear motionless to the naked eye. But in reality, Polaris is not exactly at the north celestial pole. Currently, it is about 0.75° from the north celestial pole. It is close enough to appear motionless, and bright enough to be useful as navigational aid.
It turns out that we live at the time Polaris is near the north celestial pole, but it is not always that way. The Earth experiences the precession of the equinoxes, or the axial precession. Because of this, the north and south celestial pole is slowly drifting. And consequently, Polaris itself is not always near the north celestial pole. In the future, other stars will take its place as the north star.
Precession is the same motion we observe on a rotating top. Beside the rotating motion, we can also observe the axis of rotation is also moving, and it is more evident as the top rotates slower.
In the year zero, the northern celestial pole is closer to Kochab than Polaris. But it is still too far away from the north celestial pole, so it was not usually regarded as the pole star. Pytheas —a Greek adventurer— desribed that the north celestial pole as devoid of stars.
In the next few centuries, the north celestial pole will drift too far from Polaris, and we will start calling the star Gamma Cephei as our new pole star.
Same thing is happening in south celestial pole. The closest star to the south celestial pole is currently Sigma Octantis, though it is too dim to be usable in celestial navigation, unlike Polaris in the north. In time, the south celetial pole will move away from it, and other stars will take the place as the southern pole star.
Therefore, the fact that the Earth is spherical and in motion is proven according to the condition put forward by flat-Earthers themselves. We only need a few centuries of celestial observation in order to prove it.