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Measurements of multiparticle azimuthal correlations (cumulants) for charged particles in $p$-Pb at $\sqrt{{s}_{\mathrm{NN}}}=5.02$ TeV and Pb-Pb at $\sqrt{{s}_{\mathrm{NN}}}=2.76$ TeV collisions are presented. They help address the question of whether there is evidence for global, flowlike, azimuthal correlations in the $p$-Pb system. Comparisons are made to measurements from the larger Pb-Pb system, where such evidence is established. In particular, the second harmonic two-particle cumulants are found to decrease with multiplicity, characteristic of a dominance of few-particle correlations in $p$-Pb collisions. However, when a $|\ensuremath{\Delta}\ensuremath{\eta}|$ gap is placed to suppress such correlations, the two-particle cumulants begin to rise at high multiplicity, indicating the presence of global azimuthal correlations. The Pb-Pb values are higher than the $p$-Pb values at similar multiplicities. In both systems, the second harmonic four-particle cumulants exhibit a transition from positive to negative values when the multiplicity increases. The negative values allow for a measurement of ${v}_{2}{4}$ to be made, which is found to be higher in Pb-Pb collisions at similar multiplicities. The second harmonic six-particle cumulants are also found to be higher in Pb-Pb collisions. In Pb-Pb collisions, we generally find ${v}_{2}{4}\ensuremath{\simeq}{v}_{2}{6}\ensuremath{\ne}0$ which is indicative of a Bessel-Gaussian function for the ${v}_{2}$ distribution. For very high-multiplicity Pb-Pb collisions, we observe that the four- and six-particle cumulants become consistent with 0. Finally, third harmonic two-particle cumulants in $p$-Pb and Pb-Pb are measured. These are found to be similar for overlapping multiplicities, when a $|\ensuremath{\Delta}\ensuremath{\eta}|>1.4$ gap is placed.