We investigate the dependence of steady-state properties of Schelling's segregation model on the agents' activation order. Our basic formalism is the Pollicott-Weiss version of Schelling's segregation model. Our main result modifies this baseline scenario by incorporating contagion in the decision to move: (pairs of) agents are connected by a second, agent influence network. Pair activation is specified by a random walk on this network. The considered schedulers choose the next pair nonadaptively. We can complement this result by an example of adaptive scheduler (even one that is quite fair) that is able to preclude maximal segregation. Thus scheduler nonadaptiveness seems to be required for the validity of the original result under arbitrary asynchronous scheduling. The analysis (and our result) are part of an adversarial scheduling approach we are advocating to evolutionary games and social simulations.