Renormalization-group methods developed previously to treat the growth of order in unstable systems are extended and applied to the antiferromagnetic spin-exchange (AF SE) model for order-disorder transitions in binary alloys. The number-conservation law and fixed-length sum rule are properly preserved. Various scaling behaviors are identified and the corresponding scaling functions are determined and compared to those found in previous work on the spin-flip kinetic Ising (SF KI) model. While the AF SE and SF KI models both have a nonconserved scalar order parameter, their microscopic dynamics is very different, and no quantities are conserved in the SF case. We investigate the dependence of the growth kinetics on these differences. We find that the scaling function for the quasistatic structure factor is universal for the models that we have studied. The scaling functions reflecting the dependence of the growth on the correlation length of the final equilibrium state, while quite similar for the various models, depend on both the presence of the conservation law and the choice of exchange probability. We have also carried out detailed comparisons of our results with Monte Carlo simulations for the scaling function for the structure factor, and the time dependence of the nearest-neighbor correlation function. The agreement between the theory and the simulations is excellent. In addition, we have carried out Monte Carlo simulations which verify directly the existence of the self-similar behavior on which our theory is founded.