In the interaction the Moon tries to force the two axes to coincide. This force causes the direction in which the rotation axis points to change slowly. The motion is quite regular; the path on the Celestial Sphere is roughly circular. The radius of the circle is large, ~ 23.5 degrees (is this a surprise?). The period of the motion is long, 26,000 years, due to the weakness of the effect.
Since the point where the projection of the rotation axis of the Earth strikes the Celestial Sphere moves around (the stars are still not in motion), this means that the North Celestial Pole moves around through the stars. This produces changes in the coordinates of stars (even though they are not moving). Today, it is fortuitous that the North star is Polaris. At other times, other stars will be the North star:
The Tropical year is shorter than the Sidereal year. What does this say about the direction in which the rotation axis of the Earth precesses?