Particles moving in current sheets under extreme conditions, such as those in the vicinity of pulsars or those predicted on upcoming multipetawatt laser facilities, may be subject to significant radiation losses. We present an analysis of particle motion in model fields of a relativistic neutral electron-positron current sheet in the case when radiative effects must be accounted for. In the Landau-Lifshitz radiation reaction force model, when quantum effects are negligible, an analytical solution for particle trajectories is derived. Based on this solution, for the case when quantum effects are significant an averaged quantum solution in the semiclassical approach is obtained. The applicability region of the solutions is determined and analytical trajectories are found to be in good agreement with those of numerical simulations which account for radiative effects. Based on these results we demonstrate that radiation reaction itself can provide a mechanism of pinching even within a given field consideration.