The earlier-described master equation approach to the configurational kinetics of non-equilibrium alloys is used to study kinetic features of `multivariant' orderings for which more than two types of ordered domain are possible. To this end we make extensive simulations of various phase transformations involving D0₃-type orderings for a number of alloy models. The microscopic structure of various antiphase boundaries in the D0₃ phase is also studied. A consistent approach to describing the effect of elastic forces on microstructural evolution is outlined and used to study the kinetics of a multivariant ordering accompanied by alloy decomposition. Our simulations reveal a number of peculiar kinetic features of multivariant orderings, many of them agreeing well with experimental observations.