For any local, translation-covariant quantum field theory on Minkowski spacetime we prove that two distinct primary states that are invariant under the inertial time evolutions in different inertial reference frames and satisfy a timelike cluster property called the mixing property are disjoint, i.e. each state is not a perturbation of the other. These conditions are fulfilled by the inertial KMS states of the free scalar field, thus showing that a state satisfying the KMS condition relative to one reference frame is far from thermal equilibrium relative to other frames. We review the property of return to equilibrium in open quantum systems theory and discuss the implications of disjointness on the asymptotic behavior of detector systems coupled to states of a free massless scalar field. We argue that a coupled system consisting of an Unruh-DeWitt detector moving with constant velocity relative to the field in a thermal state, or an excitation thereof, cannot approach a KMS state at late times under generic conditions. This leads to an illustration of the physical differences between heat baths in inertial systems and the apparent "heat bath" of the Unruh effect from the viewpoint of moving detectors. The article also reviews, from a quantum field theoretical perspective, the quantum dynamical system of an Unruh-DeWitt detector coupled to a massless scalar field in a KMS state relative to the rest frame of the detector.