"The equator is an imaginary line that circles the globe center. It won't go anywhere."
In a pole shift, I'm not certain that that imaginary line would mean anything anymore, hence the comment.
You're confusing the true rotational axis pole and the magnetic pole.
Not certain why that happens so often. The true north pole isn't going anywhere, hence the equator isn't going anywhere. The wanderings of the magnetic north pole have no influence on where true north is.