The β-relaxation of metallic glasses (MGs) bears nontrivial connections to their microscopic and macroscopic properties. In an effort to elucidate the mechanism of the β-relaxation, we studied by dynamical mechanical measurements the change of its properties on varying the composition of La60Ni15Al25 in various ways. The properties of the β-relaxation turn out to be very sensitive to the composition. It is found that the isochronal loss peak temperature of β-relaxation, Tβ,peak, is effectively determined by the total (La + Ni) content. When Cu is added into the alloy to replace either La, Ni, or Al, the Tβ,peak increases with decrease of the (La + Ni) content. The trend is in accordance with data of binary and ternary MGs formed from La, Ni, Al, and Cu. Binary La-Ni MGs have pronounced β-relaxation loss peaks, well separated from the α-relaxation. In contrast, the β-relaxation is not resolved in La-Al and La-Cu MGs, showing up as an excess wing. For the ternary La-Ni-Al MGs, increase of La or Ni content is crucial to lower the Tβ,peak. Keeping the Al content fixed, increase of La content lowers the Tβ,peak further, indicating the more important role La plays in lowering Tβ,peak than Ni. The observed effects on changing the composition of La60Ni15Al25 lead to the conclusion that the properties of the β-relaxation are mainly determined by the interaction between the largest solvent element, La, and the smallest element, Ni. From our data, it is further deduced that La and Ni have high mobility in the MGs, and this explains why the β-relaxation in this La-based MGs is prominent and well resolved from the α-relaxation as opposed to Pd- and Zr-based MGs where the solvent and largest atoms, Pd and Zr, are the least mobile.