The paper describes a novel technique for approximating the time-dependent neutron transport equation in the simulation of source-driven transients. The methodology, based on a physical intuition, adopts a multi-generational approach to improve the accuracy of lumped parameter approximations and better account for the propagation effects typical of a source switch-on. More precisely, the first few neutron generations are simulated collision by collision through the Boltzmann equation while a simplified lumped parameter model is adopted for the multiply collided neutrons. Numerical simulations prove that a significant gain in accuracy is obtained compared to classical point kinetics, especially for detector signal analysis. In terms of computational costs, the technique is more time consuming than point kinetics but, at low orders, remains sensibly faster than fulll time dependent transport solution or quasi-static approximations.