We study the processes in which fluctuating elements of a system are progressively fixed (quenched) while keeping the interaction with the remaining unfixed elements. If the interaction is global among Ising spin elements and if the unfixed part is reequilibrated each time after fixing an element, the evolution of a large system is martingale about the equilibrium spin value of the unfixed spins. Due to this property the system starting from the critical point yields the final magnetization, whose distribution shows non-Gaussian and slow transient behavior with the system size.