New sets of parameters (“tunes”) for the underlying-event (UE) modelling of the pythia8, pythia6 and herwig++ Monte Carlo event generators are constructed using different parton distribution functions. Combined fits to CMS UE proton–proton ( (Formula presented.) ) data at (Formula presented.) and to UE proton–antiproton ( (Formula presented.) ) data from the CDF experiment at lower (Formula presented.) , are used to study the UE models and constrain their parameters, providing thereby improved predictions for proton–proton collisions at 13 (Formula presented.). In addition, it is investigated whether the values of the parameters obtained from fits to UE observables are consistent with the values determined from fitting observables sensitive to double-parton scattering processes. Finally, comparisons are presented of the UE tunes to “minimum bias” (MB) events, multijet, and Drell–Yan ( (Formula presented.) lepton-antilepton+jets) observables at 7 and 8 (Formula presented.) , as well as predictions for MB and UE observables at 13 (Formula presented.).