Abstract: We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the parameters are such that there is an infinite component. We identify the stretch-exponent $\zeta$ of the subexponential decay of the cluster-size distribution. That is, with $|\CC(0)|$ denoting the number of vertices in the component of the vertex at $0\in \R^d$, we prove $ P (k\le |\CC(0)|<\infty)=\exp\big(-\Theta(k^{\zeta})\big), $ as $k$ tends to infinity. The value of $\zeta$ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension $d$, the power-law tail exponent $\tau$ of the degree distribution and a long-range parameter $\alpha$ governing the presence of long edges in Euclidean space.