A process of irreversible copolymerization with segmental exchange reactions is simulated using a Monte Carlo algorithm. The terminal (dyad) model is assumed for eight considered reactions: propagations and attacks of active centers on monomeric units in chains with subsequent chain scissions. It is established that microstructure of macromolecules at any given moment depends on the values of reaction rate constants and evolves from the beginning one, defined mainly by propagation rates, to the final one, determined by chain transfer rates. Dispersity and randomization degree of unit sequences in macromolecules can change nonmonotonically with progress of the process. Equilibration of the microstructure needs usually significantly longer time than needed for attaining the equilibrium chain lengths distribution. The proposed new Monte Carlo algorithm allows to more efficiently investigate the studied process.