The property of transfer between different scales of motion in evolving two-dimensional compact vortices is studied here, and a general mathematical framework is developed to describe the transfer between scales inside compact structures. This new approach is applied to the case of an axisymmetric advection which represents the leading-order (large time) approximation for Lundgren's family of two-dimensional vortices. It is also generalized to passive scalar advection by non-axisymmetric velocity fields. It is shown that scale interactions generated by an axisymmetric advection are essentially local and dominated by distant triadic interactions: in the case of an evolving spiral vortex sheet this result is confirmed even when non-axisymmetric corrections are included. A physical interpretation of the results is given, which can be summarized by saying that locality of scale interactions is caused by the uniformity of shear at a given scale and is therefore increasingly natural at small lengthscales. Local interactions are shown to arise in axisymmetric advection but to be uncommon in non-axisymmetric advection.