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. Joint work with Joost Jorritsma and Julia Komjáthy.