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Motivated by the potential use of self-rewetting fluids (i.e., fluids that exhibit a nonmonotonic variation of surface tension with temperature) in various heat-transfer applications, in the present work we formulate and analyze a theoretical model for the unsteady motion of a long bubble or droplet in a self-rewetting system in a nonuniformly heated tube due to a combination of Marangoni effects due to the variation of surface tension with temperature, gravitational effects due to the density difference between the two fluids, and an imposed background flow along the tube. We find that the evolution of the shape (but not of the position) of the bubble or droplet is driven entirely by Marangoni effects and depends on the initial value of its radius in relation to a critical value. In the case in which Marangoni effects are absent, the bubble or droplet always moves with constant velocity without changing shape. In the case in which only Marangoni effects are present, the bubble or droplet either always moves away from or always moves towards the position of minimum surface tension; in the latter case it ultimately fills the entire cross section of the tube at a final stationary position which is closer to the position of minimum surface tension than its original position. In the cases in which either only Marangoni effects and gravitational effects or only Marangoni effects and background-flow effects are present the competition between the two effects can lead to a nonmonotonic evolution of the position of the center of mass of the bubble or droplet. The behavior of a self-rewetting system described in the present work is qualitatively different from that for ordinary fluids, in which case the bubble or droplet always moves with constant velocity without changing shape.