  We consider a parabolic non-local free boundary problem that has been derived
as a limit of a bulk-surface reaction-diffusion system which models cell
polarization. In previous papers, we have established well-posedness of this
problem and derived conditions on the initial data that imply continuity of the
free boundary as $t\to 0$. In this paper we extend the qualitative study of the
free boundary by considering axisymmetric data. Under additional monotonicity
assumptions on the data we prove global continuity of the free boundary. On the
other hand, if the initial data violate a "no-fattening" condition we show that
the free boundary can oscillate as $t \to 0$.
