A number of results are established regarding the stability of the asymmetric stochastic equilibrium assignment model for general networks. In particular, we consider the marginal effect of any swap of flow from one route to an alternative, under various behavioural rules describing the way in which drivers integrate their perceived costs at equilibrium with those in disequilibrium. These pairwise route flow swaps are referred to as 'perturbations', since they assume that other route flows are held at their equilibrium levels (even though it is a non-local property, in the sense that the size of the flow swap is only limited by the demand-feasibility constraints). Loosely speaking, an equilibrium is then said to be stable if following any such perturbation, there is an incentive for drivers to re-route 'in the direction of' the same equilibrium. It is seen that if the route choice is described by a random utility model and the cost functions are monotone, then equilibrium is stable with respect to some behavioural adjustment assumptions, and unstable with respect to others. If, alternatively, the cost functions satisfy a certain derivative condition, we show that stability may be established with respect to a greater range of behavioural adjustments. By studying the limiting case as the perception error variance matrix tends to zero, it is shown that these concepts of stability may be regarded as generalisations of the 'equilibration' concept of stability proposed for deterministic equilibrium. Finally, simple examples are used to explore the application of the theorems above to study local properties of problems with multiple stochastic equilibria.