Jupiter's “gossamer” ring, which apparently consists largely of fine dust grains, and extends outward from the main ring to the vicinity of Thebe at ∼3.11 R J, has a significant peak at the synchronous radius 2.24 R J. The ring's discoverers (Showalter, Burns, Cuzzi and Pollack) believe that large bodies distributed in the vicinity of synchronous radius act as sources for this dust, which then drifts radially away from the synchronous radius, due to plasma drag. Here we will show that the so-called gyrophase drift may exceed the plasma drag drift and may move small charged grains either toward or away from synchronous radius. In the presence of a radial gradient in plasma temperature, the grain gyrophase drifts toward the higher temperature, so if the plasma is hotter at synchronous radius, the gyrophase drift could overcome the plasma drag drift and concentrate the dust at synchronous radius, with Amalthea, for example, as a source. Gyrophase drift will also take place if there is a radial gradient in the relative concentrations of different plasma ion species, or because of plasma-grain velocity variation due to the grain's cycloidal motion through the plasma. In order to popularize the gyrophase drifts as ones which must be taken into account in any study of the fine grain dynamics in a magnetosphere, we evaluate some of them for grains launched at the local Kepler velocity. We also display numerical orbit integrations for comparison with the analytic adiabatic theory. The Poynting-Robertson drift is evaluated and found to be miniscule compared to the plasma drag or gyrophase drifts. The inner edge of the Io plasma torus is the location of a very steep gradient in plasma temperature which may, by means of the gyrophase drift, act to keep fine dust from Io's volcanos from penetrating to the inner magnetosphere. However, there is diffusive motion superposed on the steady gyrophase drift. For sufficiently large dust grains this diffusion will overcome the excluding effect of gyrophase drift and permit these grains to reach the inner magnetosphere.