We consider the Dirichlet problem for linear nonautonomous second order parabolic equations  with bounded measurable coefficients on bounded Lipschitz domains. Using a new  Harnack-type inequality for  quotients of positive solutions, we show that each positive solution exponentially dominates any solution which changes sign for all times. We then examine continuity and robustness properties of  a principal Floquet bundle and the associated exponential separation under perturbations of the coefficients and the spatial domain.