Measurements of strange hadron (K0s, Lambda + anti-Lambda, and Xi+ + Xi-) transverse momentum spectra in pp, pPb, and PbPb collisions are presented over a wide range of rapidity and event charged-particle multiplicity. The data were collected with the CMS detector at the CERN LHC in pp collisions at sqrt(s) = 7 TeV, pPb collisions at sqrt(s[NN]) = 5.02 TeV, and PbPb collisions at sqrt(s[NN]) = 2.76 TeV. The average transverse kinetic energy is found to increase with multiplicity, at a faster rate for heavier strange particle species in all systems. At similar multiplicities, the difference in average transverse kinetic energy between different particle species is observed to be larger for pp and pPb events than for PbPb events. In pPb collisions, the average transverse kinetic energy is found to be slightly larger in the Pb-going direction than in the p-going direction for events with large multiplicity. The spectra are compared to models motivated by hydrodynamics.