We analyse the Lagrangian geometry of a time-dependent flow in the neighbourhood of a coherent vortex which is assumed to be a member of a vortex cluster collectively ruling the dynamics in the late stages of decaying two-dimensional turbulence. In order to gain insight into the highly inhomogeneous transport of a passive tracer in such vortex-dominated flows, we consider an idealised kinematic model of a local flow around a single coherent vortex in the cluster at distances much smaller than the distance to its nearest neighbour. We focus, in particular, on flow configurations which lead to a vigorous stirring and a subsequent escape of a passive tracer from the neighbourhood of the vortex. Here, the central vortex is approximated by a point vortex but the analytical arguments can be modified to cater for more realistic vorticity distributions. The principal axes of an irrotational ambient strain, which represents the combined, leading-order influence of its neighbours, are assumed to rotate with constant angular velocity and the strain-rate varies harmonically in time. The Lagrangian structure of the flow near the vortex is analysed by utilising the Hamiltonian formalism and employing appropriate perturbation methods. It is shown that sufficiently near the vortex there exist KAM-like tori which confine regions of purely chaotic tracer trajectories to the neighbourhood of the vortex. We emphasise, however, that there can exist certain 'open' flow geometries which lead to eventual 'leakage' of the tracer from 'sufficiently distant' regions of vigorous stirring to the outer flow. Such local flow configurations can be regarded as a prototype of the 'mixers' in decaying 2D turbulence.