Measurements of two-and multi-particle angular correlations in pp collisions at √s = 5, 7,and13 TeVare presented as a function of charged-particle multiplicity. The data, corresponding to integrated luminosities of 1.0 pb⁻¹ (5TeV), 6.2 pb⁻¹ (7TeV), and 0.7 pb⁻¹(13 TeV), were collected using the CMS detector at the LHC. The second-order (v₂) and third-order (v₃) azimuthal anisotropy harmonics of unidentified charged particles, as well as v₂ of K⁰_S and Λ/Λ¯ particles, are extracted from long-range two-particle correlations as functions of particle multiplicity and transverse momentum. For high-multiplicity pp events, a mass ordering is observed for the v₂ values of charged hadrons (mostly pions), K⁰_S, and Λ/Λ¯, with lighter particle species exhibiting a stronger azimuthal anisotropy signal below p_T ≈ 2 GeV/c. For 13 TeV data, the v₂ signals are also extracted from four-and six-particle correlations for the first time in pp collisions, with comparable magnitude to those from two-particle correlations. These observations are similar to those seen in pPb and PbPb collisions, and support the interpretation of a collective origin for the observed long-range correlations in high-multiplicity pp collisions.