Time-dependent Monte Carlo simulations for reactor kinetics applications require special variance-reduction and population-control techniques in order to efficiently cope with the typically huge imbalance between the respective time scales and population sizes of neutrons and precursors. Building upon the legacy implementation of the algorithms devoted to kinetics in the Monte Carlo code TRIPOLI-4, in this work we propose an adaptive adjoint-based population-control method that considerably improves the behaviour of time-dependent simulations. Thanks to a time-dependent importance-sampling scheme, based on the solution of the adjoint point-kinetics equations, neutron and precursor weights are continuously adjusted, which paves the way towards the simulation of previously unattainable reactor transients involving long times and large reactivity excursions. The computational effectiveness of the newly developed method is evaluated in terms of Figure of Merit (FoM) over a set of time-dependent scenarios encompassing the Flattop-Pu, SPERT III E-core and CROCUS benchmarks.