Imagine we stop the earth's rotation, just to illustrate how objects moving towards east or west behave.
Imagine you could shoot a magic bullet, one that keeps moving forever at the same speed and height above earth's surface.
If you shoot the bullet towards the east, it will NOT circle the earth at the same latitude. It will start going towards equator.
The reason is that all such movement will follow a great circle. Hence, there is always a north / south movement, unless you are on the equator.
Imagine standing on the north pole. You walk a hundred yards in any direction (south). You shoot your magic bullet to the east. You wouldn't expect it to circle the pole, and hit you in the back a second later, no ? It will fly all the way towards the south pole, pass it at a hundred yards, and then back.