I found this question in the back of my DE book. I can't think of anything that applies.
Give an example of a nonautonomous DE for which there is a solution $y$ with $y(t) \to q$ as $t \to \infty$, but $q$ is not an equilibrium point. You do not need to think beyond a first order linear homogeneous equation $y'(t)=a(t)y$.