Oscillating electrowetting on dielectrics (EWOD) with coplanar electrodes is investigated in this paper as a way to provide efficient stirring within a drop with biological content. A supporting model inspired from Ko et al. [Appl. Phys. Lett. 94, 194102 (2009)] is proposed allowing to interpret oscillating EWOD-induced drop internal flow as the result of a current streaming along the drop surface deformed by capillary waves. Current streaming behaves essentially as a surface flow generator and the momentum it sustains within the (viscous) drop is even more significant as the surface to volume ratio is small. With the circular electrode pair considered in this paper, oscillating EWOD sustains toroidal vortical flows when the experiments are conducted with aqueous drops in air as ambient phase. But when oil is used as ambient phase, it is demonstrated that the presence of an electrode gap is responsible for a change in drop shape: a pinch-off at the electrode gap yields a peanut-shaped drop and a symmetry break-up of the EWOD-induced flow pattern. Viscosity of oil is also responsible for promoting an efficient damping of the capillary waves which populate the surface of the actuated drop. As a result, the capillary network switches from one standing wave to two superimposed traveling waves of different mechanical energy, provided that actuation frequency is large enough, for instance, as large as the one commonly used in electrowetting applications (f $ 500 Hz and beyond). Special emphasis is put on stirring of biological samples. As a typical application, it is demonstrated how beads or cell clusters can be focused under flow either at mid-height of the drop or near the wetting plane, depending on how the nature of the capillary waves is (standing or traveling), and therefore, depending on the actuation frequency (150 Hz-1 KHz).