We study the kink motion for the one-dimensional stochastic Allen-Cahn equation and its mass conserving counterpart. Using a deterministic slow manifold, in the sharp interface limit for sufficiently small noise strength we derive an explicit stochastic differential equation for the motion of the interfaces, which is valid as long as the solution stays close to the manifold. On a relevant time-scale, where interfaces move at most by the minimal allowed distance between interfaces, we show that the kinks behave approximately like the driving Wiener-process projected onto the slow manifold, while in the mass-conserving case they are additionally coupled via the mass constraint.