Aims. The aim of this work is to investigate in a physical and quantitative way the spectral evolution of bright neutron star low-mass X-ray binaries (NS LMXBs) with special regard to the transient hard X-ray tails. Methods. We analyzed INTEGRAL data for five sources (GX 5-1, GX 349+2, GX 13+1, GX 3+1, GX 9+1) and built broad-band X-ray spectra from JEM-X1 and IBIS/ISGRI data. For each source, X-ray spectra from different states were fitted with the recently proposed model compTB. Results. The spectra have been fit with a two-compTB model. In all cases the first compTB describes the dominant part of the spectrum that we interpret as thermal Comptonization of soft seed photons (<1 keV), likely from the accretion disk, by a 3−5 keV corona. In all cases, this component does not evolve much in terms of Comptonization efficiency, with the system converging to thermal equilibrium for an increasing accretion rate. The second compTB varies more dramatically, spanning from bulk plus thermal Comptonization of blackbody seed photons to the blackbody emission alone. These seed photons (R < 12 km, kT s > 1 keV), likely from the neutron star and the innermost part of the system, the transition layer, are Comptonized by matter in a converging flow. The presence and nature of this second compTB component (whether a pure blackbody or Comptonized) are related to the inner local accretion rate which can influence the transient behavior of the hard tail: high values of accretion rates correspond to an efficient bulk Comptonization process (bulk parameter δ 0), while even higher values of accretion rates suppress the Comptonization, resulting in simple blackbody emission (δ = 0). Conclusions. The spectral evolution of the sources has been successfully studied in terms of thermal and bulk Comptonization efficiency in relation to the physical conditions in the transition layer.