For a binary alloy with an ordered structure B2, we research the atomic migration by means of kinetic Monte Carlo simulations, where the atomic migration results from the exchange of positions between an atom and a vacancy in a rigid lattice. The atomistic kinetic model we used is based on the jump rate theory and the residence time algorithm, wherein we consider the pair interactions between neighbours up to the next-nearest neighbours. Likewise, we determine the ratio of the diffusion coefficients as a function of temperature and investigate the antiphase boundary mobility in the last stages of the ordering process. Furthermore, we calculate the autocorrelation function, which reveals that, in the lattice, not only the vacancy makes highly correlated jumps at low temperatures but the atoms jump to positions of their sub-lattice at moderate temperatures.