Results of computer simulations of the kinetics of correlated annealing of pairs of close a-I and F-I centers in KCI and KBr crystals, enhanced by I-center diffusion and Coulomb or elastic attractions, respectively, are presented. Special attention is paid to the conditions under which multi-stage annealing stages arise as it has been observed experimentally more than once. Our general conclusions are: (i) a weak elastic interaction affects the recombination kinetics and the survival probability even for relatively well-separated F-I pairs, the more so is true for the case of Coulomb attraction between charged a-I pairs; (ii) the multi-step (kink) structure arises only for close (typically, up to fourth-nearest neighbour) defects. To explain the strongly separated stages observed experimentally for a-I (17 to 21 K, 30 to 39 K in KCI and 18 to 22 K and 27 to 30 K in KBr) a very particular initial distribution of defects is required consisting of two groups of close and well-separated defects. It gives strong support to the results of recent computer simulations that an interstitial ion created due to electron trapping by an H center can be transformed into a crowdion which could be displaced athermally by a long distance from a vacancy (five to seven interionic distances).