A study is presented of B+→KS0K-π+K+ and B+→KS0K+π-K+ decays based on the analysis of proton-proton collision data collected with the LHCb detector at center-of-mass energies of 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb-1. The KS0Kπ invariant-mass distributions of both B+ decay modes show, in the m(KS0Kπ)<1.85 GeV mass region, large activity which is resolved using an amplitude analysis. A simple model, where JPC amplitudes are described by multiple Breit-Wigner functions with appropriate angular distributions, provides a good description of the experimental data. In this approach a complex mixture of JPC=0-+, 1++ and 1+- amplitudes is observed that is dominated by η(1405), η(1470), η(1760), f1(1285), f1(1420) and h1(1405) resonances. The KS0Kπ Dalitz plots are dominated by asymmetric crossing K∗K̄ bands which are different for the two B+ decay modes. This is due to a different interference pattern between the 1++ and 1+- amplitudes in the two channels. Branching fractions are measured for each resonant contribution.