Lagrangian coherent sets are known to crucially determine transport and mixing processesin non-autonomous flows. Prominent examples include vortices and jets in geophysical fluid flows.Coherent sets can be identified computationally by a probabilistic transfer-operator-based approachwithin a set-oriented numerical framework. Here, we study sudden changes in flow patternsthat correspond to bifurcations of coherent sets. Significant changes in the spectral properties ofa numerical transfer operator are heuristically related to critical events in the phase space of atime-dependent system. The transfer operator approach is applied to different example systems ofincreasing complexity. In particular, we study the 2002 splitting event of the Antarctic polar vorte