We used Monte Carlo simulations to study the kinetics and steady states of B2 and B32 ordering in an equiatomic binary bcc alloy having both thermal and ballistic atom movements. Atom movements occurred by a vacancy mechanism, where first-neighbor atoms exchanged sites with the vacancy. In the thermodynamic case, where this jump probability was set by a Boltzmann factor alone, we found the formation of large amounts of transient B2 (B32) during disorder → B32 (B2) order transformations. The transient order developed in distinct regions, but vanished completely when equilibrium was attained. Ballistic jumps were included as random interchanges of the vacancy with one of its neighboring atoms. Even with small fractions of ballistic jumps, there were large changes in the transient states and steady states of order in the alloy. A kinetic explanation is proposed, in which the presence of ballistic jumps contributed a greater amount of internal energy to the B2 phase than to the B32 phase when there was a strong asymmetry in the second-nearest neighbor pair potentials (VAA2 ↗ VBB2). A two-phase coexistence in the steady state was explained by fluctuations in the local density of ballistic jumps.