We study the propagation of nonrelativistic bound states moving at constant velocity across a homogeneous thermal bath and we develop the effective field theory which is relevant in various dynamical regimes. We consider values of the velocity of the bound state ranging from moderate to highly relativistic and temperatures at all relevant scales smaller than the mass of the particles that form the bound state. In particular, we consider two distinct temperature regimes, corresponding to temperatures smaller or higher than the typical momentum transfer in the bound state. For temperatures smaller or of the order of the typical momentum transfer, we restrict our analysis to the simplest system, a hydrogenlike atom. We build the effective theory for this system first considering moderate values of the velocity and then the relativistic case. For large values of the velocity of the bound state, the separation of scales is such that the corresponding effective theory resembles the soft collinear effective theory (SCET). For temperatures larger than the typical momentum transfer we also consider muonic hydrogen propagating in a plasma which contains photons and massless electrons and positrons, so that the system resembles very much a heavy quarkonium in a thermal medium of deconfined quarks and gluons. We study the behavior of the real and imaginary part of the static two-body potential, for various velocities of the bound state, in the hard thermal loop approximation. We find that Landau damping ceases to be the relevant mechanism for dissociation from a certain critical velocity on, in favor of screening. Our results are relevant for understanding how the properties of heavy quarkonia states produced in the initial fusion of partons in the relativistic collision of heavy ions are affected by the presence of an equilibrated quark-gluon plasma.