Using direct numerical simulations, it is established that under realistic conditions, the turbulent transport in magnetized fusion plasmas tends to be in a regime for which the cross-field dynamics of tracers can be described neither by Eulerian nor by random walk approaches. Instead, the concept of Lagrangian coherent structures turns out to be a useful tool for studying such systems. Here, networks of repelling and attracting material lines-acting as barriers for transport - become important. The latter are analogues of the stable and unstable manifolds in the static case. This opens up new possibilities for interpreting and analyzing turbulent transport in magnetized plasmas.