I will consider a simple population model in which individuals produce offspring independently of each other according to a general offspring distribution, and die at a rate which depends on the total population size. In such models, we can use neutral markers to keep track of the dynamics of the relative sizes of several subfamilies. I will be interested in the limiting behaviour of these neutral markers when the population size tends to infinity in a weak competition regime. This will yield three possible regimes depending on the tail of the offspring distribution. In each regime, the limiting dynamics of neutral markers is given by a so-called Lambda-Fleming-Viot process, as introduced by Bertoin and Le Gall in 2003, whose Lambda measure depends on the offspring distribution. These results are in direct correspondence with those of Schweinsberg (2003) who showed that the genealogies of samples of individuals in some Cannings models converge to Lambda coalescents as the population size tends to infinity. One novelty here is that the population size is not fixed a priori and fluctuates randomly around some large value.