We study the manipulation of Majorana zero modes in a thin disk made from a $p$-wave superconductor, in order to understand their use as a building block for topological quantum computers. We analyze the second-order topological corner modes that arise when an in-plane magnetic field is applied, and calculate their dynamical evolution when rotating the magnetic field, with special emphasis on non-adiabatic effects. We characterize the phase transition between high-frequency and near-adiabatic evolution using Floquet analysis. We show that oscillations persist even in the adiabatic phase because of a frequency dependent coupling between zero modes and excited states, which we have quantified numerically and analytically. These results show that controlling the rotation frequency can be a simple method to avoid the non-adiabatic errors originated from this coupling and thus increase the robustness of topological quantum computation.
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