During the meandering motion of a dipolar vortex on a ß-plane mass is exchanged both between the dipole and the ambient fluid and between the two dipole halves. The mass exchange (as well as the meandering motion) is caused by variations of the relative vorticity of the vortices due to conservation of potential vorticity. Previous studies have shown that a modulated point-vortex model captures the essential features in the dipole evolution. For this model we write the equations of motion of passive tracers in the form of a periodically perturbed integrable Hamiltonian system and subsequently study transport using a ‘dynamical-systems theory’ approach. The amount of mass exchanged between different regions of the flow is evaluated as a function of two parameters: the gradient of ambient vorticity, ß, and the initial direction of propagation of the dipole, a0. Mass exchange between the dipole and the surroundings increases with increasing both ß and a0. The exchange rate (amount of mass exchanged per unit time) increases with ß and has a maximum for a particular value of a0 ([approximate] 0.62p). Dipolar vortices in a rotating fluid (with a sloping bottom providing the ‘topographic’ ß-effect) show, in addition to the relative vorticity variations, a second perturbation that leads to exchange of mass. The points where vorticity is extreme approach each other as the dipole moves to shallower parts of the fluid and separate as the couple moves to deeper parts. This mechanism is studied independently and it is shown to lead to a stronger exchange between the dipole halves and the ambient fluid but no exchange between the two dipole halves.