We propose a general method to calculate the drift velocity of cholesteric textures subjected to a temperature gradient when the backflow effects are negligible. The textures may be Translationally Invariant Configurations (TICs) or localized structures such as cholesteric droplets or cholesteric fingers. For the TICs and for the droplets, the drift is rotational while for the fingers, the drift is translational. We show that for the TICs, the drift is only due to the thermomechanical coupling terms of Leslie (classical term) and of Akopyan and Zel’dovich (which are additional texture-dependent terms). For the localized structures, we show that another mechanism involving the temperature variations of the elastic constants and the existence of a transverse temperature gradient can lead to a drift which adds to the one due the classical thermomechanical effects.