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We investigate the transverse momentum ($p_{\rm T}$) spectra of identified hadrons in minimum-bias proton-proton (pp) collisions at a centre-of-mass energy ($\sqrt{s}$) of 0.9, 2.76, 5.02, 7 and 13 TeV in the framework of Tsallis-blast wave (TBW) model. It is found that the model describes well the particle spectra up to 10 GeV/c. The radial flow ($\langle β\rangle$) increases with the collision energy. The degrees of non-equilibrium ($q$) and the Tsallis temperature parameter ($T$) show a similar behaviour, but with a much weaker trend. With this dependence of the freeze-out parameters on the collision energy, we evaluate $\langle β\rangle$, $T$ and $q$ in pp collisions at $\sqrt{s}=$ 8 and 14 TeV and predict the particle spectra at these two energies. Moreover, in order to investigate the multiplicity dependence of the freeze-out parameters, the TBW model is extended to the spectra at different charged-particle multiplicity classes in pp collisions at $\sqrt{s}=$ 7 and 13 TeV. It is observed that at both energies the radial flow increases with the multiplicity while the degree of non-equilibrium shows an opposite behaviour, which is similar to that observed in proton-nucleus (pA) and nucleus-nucleus (AA) collisions at the Large Hadron Collider (LHC) energies. However, the Tsallis temperature parameter increases with the multiplicity, which is opposite to the trend in pA and AA collisions. At similar multiplicities, the radial flow in pp collisions is stronger than those in pA and AA collisions, indicating that the size of the colliding system has significant effects on the final state particle dynamics. Finally, we apply an additional flow correction to the Tsallis temperature parameter and find that the doppler-corrected temperature parameter almost scales with the multiplicity in a uniform way, despite the difference in the colliding system and collision energy.