Traditional human societies are organised around kinship, and use kinship networks to generate large scale community projects. This is made possible by a combination of linguistic kin recognition, a uniquely human trait, which is mediated by the reliability of kin as collaborators. When effective fertility falls, this results in two simultaneous effects on social networks: there are fewer kin that can be relied on, and the limiting effect of the local kin-clustering becomes stronger. To capture this phenomenon, we used a model of kinship lineages to build populations with a range of fertility levels combined with a behavioural synchrony model to measure the efficiency of collective action generated on kin networks within populations. Our findings suggest that, whenever effective cooperation depends on kinship, falling fertility creates a crisis when it results in too few kin to join the community project. We conclude that, when societies transition to small effective kin networks, due to falling fertility, increased relative distance to kin due to urbanisation or high mortality due to war or epidemics, they will be able to remain socially cohesive only if they replace disappearing kin networks with quasi-kin alternatives based on membership of guilds or clubs.