The "out of the blue" nature of recent terror attacks and the diversity of apparent motives highlight the importance of understanding the online trajectories that individuals follow prior to developing high levels of extremist support. Here we show that the physics of stochastic walks, with and without temporal correlation, provides a unifying description of these online trajectories. Our unique data set comprising all users of a global social media site reveals universal characteristics in individuals' online lifetimes. Our accompanying theory generates analytical and numerical solutions that describe the characteristics shown by individuals that go on to develop high levels of extremist support and those that do not. Going forward, it is conceivable that a deeper understanding of these temporal and many-body correlations may eventually contribute to the important task of better countering the spread of radical propaganda online.