  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.
