Multi-particle cumulants and corresponding Fourier harmonics are measured for azimuthal angle distributions of charged particles in $$pp$$ collisions at $$\sqrt{s}$$ = 5.02 and 13 TeV and in $$p$$ + Pb collisions at $$\sqrt{s_{_\text {NN}}}$$ = 5.02 TeV, and compared to the results obtained for low-multiplicity $$\mathrm{Pb}~+~\mathrm{Pb}$$ collisions at $$\sqrt{s_{_\text {NN}}}$$ = 2.76 TeV. These measurements aim to assess the collective nature of particle production. The measurements of multi-particle cumulants confirm the evidence for collective phenomena in $$p$$ + Pb and low-multiplicity $$\mathrm{Pb}~+~\mathrm{Pb}$$ collisions. On the other hand, the $$pp$$ results for four-particle cumulants do not demonstrate collective behaviour, indicating that they may be biased by contributions from non-flow correlations. A comparison of multi-particle cumulants and derived Fourier harmonics across different collision systems is presented as a function of the charged-particle multiplicity. For a given multiplicity, the measured Fourier harmonics are largest in $$\mathrm{Pb}~+~\mathrm{Pb}$$ , smaller in $$p$$ + Pb and smallest in $$pp$$ collisions. The $$pp$$ results show no dependence on the collision energy, nor on the multiplicity.