I use multitype branching processes to study genetic models for the evolution of social behaviour, i.e. behaviours that, when acted out, affect the success of the actor's neighbours. Here, I suppose an individual bearing a mutant copy of a gene influences the reproductive success of a neighbour by altering its own competitive ability. Approximations based on assumptions about the rareness of the mutant allele and the strength of selection allow me to formulate statements concerning the probability of mutant extinction in terms of inclusive fitness. Inclusive fitness is an idea well known to biologists and can be thought of as a sum of an individual's fitness and the fitness of each of its relatives, weighted by some measure of genetic relatedness. Previous work has led to some confusion surrounding the definition of the inclusive-fitness effect of a mutant allele when individuals carrying that allele experience demographic conditions that fluctuate randomly. In this paper, I emphasise the link between inclusive fitness and the probability of mutant extinction. I recover standard results for populations of constant size, and I show that inclusive fitness can be used to determine the short-term fate of mutants in the face of stochastic demographic fluctuations. Overall, then, I provide a connection between certain inclusive-fitness-based approaches routinely applied in theoretical studies of social evolution.