  We investigate within the Darmois-Israel thin shell formalism the match of
neutral and asymptotically flat, slowly rotating spacetimes (up to the second
order in the rotation parameter) when their boundaries are dynamic. It has
several important applications in general relativistic systems, such as black
holes and neutron stars, which we exemplify. We mostly focus on stability
aspects of slowly rotating thin shells in equilibrium and surface degrees of
freedom on the hypersurfaces splitting the matched slowly rotating spacetimes,
e.g., surface energy density and surface tension. We show that the stability
upon perturbations in the spherically symmetric case automatically implies
stability in the slow rotation case. In addition, we show that when matching
slowly rotating Kerr spacetimes through thin shells in equilibrium, surface
degrees of freedom can decrease compared to their Schwarzschild counterparts,
meaning that energy conditions could be weakened. Frame-dragging aspects of the
match of slowly rotating spacetimes are also briefly discussed.
