We study the dynamics of mixing of advected fields in open chaotic flows. We propose the eigenfunctions of the stroboscopic advection-diffusion (SAD) operator as the natural generalisations of the concept of strange eigenmodes for open flows, and argue that its eigenvalues determine the long-time dynamics of mixing. We characterise their dependence on diffusivity and on the properties of the chaotic advection. In particular, we find that the SAD eigenvalues are determined by the dynamical invariants of the chaotic saddle, and that the eigenmodes mirror its fractal geometry. Furthermore, we find that the dependence of the SAD eigenvalues on the diffusivity is strikingly different for hyperbolic and non-hyperbolic flows. In the latter case, we show strong evidence of an anomalous scaling of the eigenvalues with the diffusivity.