We investigate the stellar-mass Tully-Fisher relation (TFR) between the stellar mass and the integrated gas velocity dispersion, quantified by the kinematic estimator S_0.5 measured from strong emission lines in spectra of galaxies at 0<z<5. We combine luminosity-selected galaxies (`high-luminosity sample) with galaxies selected in other ways (`low-luminosity sample) to cover a range in stellar mass that spans almost five orders of magnitude: 7.0 < log M* < 11.5. We find that the logarithmic power-law slope and normalisation of the TFR are independent of redshift out to z~3. The scatter in the TFR is <0.5 dex such that the gas velocity dispersion can be used as a proxy for the stellar mass of a galaxy independently of its redshift. At z>3 the scatter increases and the existence of a correlation is not obvious. The high-luminosity sample exhibits a flatter slope of 1.5$pm$0.2 at z<3 compared to the low-luminosity sample slope of 2.9$pm$0.3, suggesting a turnover in the TFR. The combined sample is well fit with a break in the TFR at a characteristic stellar mass scale of M*~10$^{10}$ M$_{odot}$, with no significant evolution out to z~3. We demonstrate that a break in the TFR with a steeper slope at the low-mass end is a natural consequence of galaxy models with a mass-dependent stellar to halo-mass ratio.
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