We consider the use of a kinetic Monte Carlo approach for the description of nonequilibrium bosonic systems, taking nonresonantly excited exciton-polariton condensates and bosonic cascade lasers as examples. In the former case, the considered approach allows the study of the cross-over between incoherent and coherent regimes, which represents the formation of a quasicondensate that forms purely from the action of energy relaxation processes rather than interactions between the condensing particles themselves. In the latter case, we show theoretically that a bosonic cascade can develop an output coherent state.