Chaotic advection of non-diffusing Lagrangian particles is often studied for model flows in the absence of molecular diffusion, which leaves open the questions of (i) whether any such flows are physically realizable, and if they are, (ii) how are transport rates affected by the mixing? To address these issues, we consider transport of heat or mass from circulating droplets that are both settling and subject to a time-dependent axial electric field. The oscillatory electric field drives an electrohydrodynamic flow which augments the Hadamard circulation caused by the steady translation, and which results in chaotic advection within the drop. The problem is governed by four parameters: the Peclet number, the dimensionless amplitudes of both the steady and oscillatory parts of the electric field, and the dimensionless frequency of the modulation. The convective diffusion equation is solved numerically for a wide range of these parameters. The results are characterized by the asymptotic rate of extraction of heat and/or mass from the droplet, (which is found to be exponential in time), and the enhancement of the transport rates is studied as a function of parameters. Somewhat surprisingly, the enhancement is not a monotonic function of the frequency but rather, exhibits spectral 'resonant peaks' at particular values of the frequency, leading to very significant increases in the extraction rate. Scientific visualizations are used to determine that there exist underlying time-periodic spatial structures of the concentration field, so called "strange eigenmodes", and that the temporal phase relationship between lobes of these eigenmodes is responsible for the resonant behavior. If time allows, extension to three-dimensional flows will be mentioned.