Measurements of strange hadron (View the MathML sourceKS0, View the MathML sourceΛ+Λ‾, and View the MathML sourceΞ−+Ξ‾+) transverse momentum spectra in pppp, pPbpPb, 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 pppp collisions at View the MathML sources=7TeV, pPbpPb collisions at View the MathML sourcesNN=5.02TeV, and PbPb collisions at View the MathML sourcesNN=2.76TeV. 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 pppp and pPbpPb events than for PbPb events. In pPbpPb 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.