Summary: This article presents a variant of Fleming-Viot particle systems, which are a standard way to approximate the law of a Markov process with killing as well as related quantities. Classical Fleming-Viot particle systems proceed by simulating \(N\) trajectories, or particles, according to the dynamics of the underlying process, until one of them is killed. At this killing time, the particle is instantaneously branched on one of the \((N-1)\) other ones, and so on until a fixed and finite final time \(T\). In our variant, we propose to wait until \(K\) particles are killed and then rebranch them independently on the \((N-K)\) alive ones. Specifically, we focus our attention on the large population limit and the regime where \(K/N\) has a given limit when \(N\) goes to infinity. In this context, we establish consistency and asymptotic normality results. The variant we propose is motivated by applications in rare event estimation problems through its connection with Adaptive Multilevel Splitting and Subset Simulation.