We consider the contribution of the solar wind-driven Dungey-cycle to flux transport in Jupiter's and Saturn's magnetospheres, the associated voltages being based on estimates of the magnetopause reconnection rates recently derived from observations of the interplanetary medium in the vicinity of the corresponding planetary orbits. At Jupiter, the reconnection voltages are estimated to be ~150 kV during several-day weak-field rarefaction regions, increasing to ~1 MV during few-day strong-field compression regions. The corresponding values at Saturn are ~25 kV for rarefaction regions, increasing to ~150 kV for compressions. These values are compared with the voltages associated with the flows driven by planetary rotation. Estimates of the rotational flux transport in the "middle" and "outer" magnetosphere regions are shown to yield voltages of several MV and several hundred kV at Jupiter and Saturn respectively, thus being of the same order as the estimated peak Dungey-cycle voltages. We conclude that under such circumstances the Dungey-cycle "return" flow will make a significant contribution to the flux transport in the outer magnetospheric regions. The "return" Dungey-cycle flows are then expected to form layers which are a few planetary radii wide inside the dawn and morning magnetopause. In the absence of significant cross-field plasma diffusion, these layers will be characterized by the presence of hot light ions originating from either the planetary ionosphere or the solar wind, while the inner layers associated with the Vasyliunas-cycle and middle magnetosphere transport will be dominated by hot heavy ions originating from internal moon/ring plasma sources. The temperature of these ions is estimated to be of the order of a few keV at Saturn and a few tens of keV at Jupiter, in both layers.