Summary: We try to carry over, as closely as possible, the well-known results for rotational dragging (Thirring, Brill and Cohen) to dragging due to linearly accelerated masses. To this end, a spherical, charged mass shell is linearly accelerated by a (weak) external, axisymmetric and dipolar charge distribution. It is shown that the interior of this (Reissner--Nordström-like) shell stays flat. The dragging of neutral test particles inside the shell, defined by their acceleration, scaled by the overall acceleration of the (rigid) shell, is calculated for the weak field case for a highly massive but weakly charged shell and for the general strong field case. The results compare favourably with the corresponding results for rotational dragging.