Can the validity of time-independent perturbation theory wavefunctions decrease over time?

In time independent perturbation theory one obtains an approximation $psi^{(1)}$ to the actual energy eigenstates $psi$ of the new system.

However since this approximation is not itself a stationary state in the new system, it will evolve over time. Does this mean that, in the Hilbert space, the discrepancy between the approximated eigenstate and the actual eigenstate could increase and decrease over time since the values of $psi^{(1)}(t)$ in state space will change over time and could get closer to the actual eigenstate $psi(t)$ which evolves only by a phase?