Galaxy mergers are believed to trigger strong starbursts. This is well assessed by observations in the local Universe. However, the efficiency of this mechanism has poorly been tested so far for high-redshift, actively star-forming, galaxies. We present a suite of pc-resolution hydrodynamical numerical simulations to compare the star formation process along a merging sequence of high- and low-redshift galaxies, by varying the gas mass fraction between the two models. We show that, for the same orbit, high-redshift gas-rich mergers are less efficient than low-redshift ones at producing starbursts; the star formation rate excess induced by the merger and its duration are both around 10 times lower than in the low gas fraction case. The mechanisms that account for the star formation triggering at low redshift - the increased compressive turbulence, gas fragmentation, and central gas inflows - are only mildly, if not at all, enhanced for high gas fraction galaxy encounters. Furthermore, we show that the strong stellar feedback from the initially high star formation rate in high-redshift galaxies does not prevent an increase of the star formation during the merger. Our results are consistent with the observed increase of the number of major mergers with increasing redshift being faster than the respective increase in the number of starburst galaxies.