In order to `zoom in' on a potential Navier-Stokes singularity, it is natural to consider sequences of Navier-Stokes solutions whose initial data are converging only in a weak-* sense. We identify a natural class of solutions satisfying the following stability property: weak-* convergence of the initial data in critical Besov spaces implies strong convergence of the corresponding solutions. We present applications of the weak-* stability property to problems concerning blow-up criteria in critical spaces, minimal blow-up initial data, and forward self-similar solutions. Finally, we discuss various difficulties concerning the analogous problem in BMO-1. Joint work with Tobias Barker (ENS).