  We find two types of moving dark soliton textures in elongated Bose-Einstein
condensates: non-stationary kinks and proper dark solitons. The former have a
curved notch region and rapidly decay by emitting phonons and/or proper dark
solitons. The proper moving solitons are characterized by a flat notch region
and we obtain the diagram of their dynamical stability. At finite temperatures
the dynamically stable solitons decay due to the thermodynamic instability. We
develop a theory of their dissipative dynamics and explain experimental data.
