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We study the physics of the quantum spin liquid (QSL) candidate Ce₂Zr₂O₇ whose magnetic properties emerge from the dipole-octupole nature of the magnetic cerium ions. Its low-energy description is in terms of an effective pseudo spin-1/2 Hamiltonian where only the local z-component of the applied field linearly couples to the local x- and z-component of the spin while the octupolar y-component remains invisible. Using the Hamiltonian we obtained in our previous study [A. Bhardwaj et al.,arXiv:2108.01096] which captures the dynamical features in zero field, we extend our analysis to the case of finite applied magnetic fields. For this purpose, we perform a combination of classical Monte Carlo and Landau-Lifshitz dynamical calculations and Lanczos both at zero and finite temperature. We find that the continuum seen in the dynamical structure factor, consistent with the existence of a gapless QSL, is largely suppressed on the introduction of a magnetic field, giving way to Bragg peaks. However, the absence of any dispersive modes is strongly reflective of the octupolar nature of the low energy modes, a finding that can be directly tested in neutron experiments.