Galaxy clusters are unique laboratories for studying astrophysical processes and their impact on halo gas kinematics. Despite their importance, the full complexity of gas motion within and around these clusters remains poorly known. This paper is part of a series presenting the first results from the new TNG-Cluster simulation, a suite comprising 352 high-mass galaxy clusters including the full cosmological context, mergers and accretion, baryonic processes and feedback, and magnetic fields. Studying the dynamics and coherence of gas flows, we find that gas motions in galaxy cluster cores and intermediate regions are largely balanced between inflows and outflows, exhibiting a Gaussian distribution centered at zero velocity. In the outskirts, even the net velocity distribution becomes asymmetric, featuring a double peak where the second peak reflects cosmic accretion. Across all cluster regions, the resulting net flow distribution reveals complex gas dynamics. These are strongly correlated with halo properties: at a given total cluster mass, unrelaxed, late-forming halos with fewer massive black holes and lower accretion rates exhibit a more dynamic behavior. Our analysis shows no clear relationship between line-of-sight and radial gas velocities, suggesting that line-of-sight velocity alone is insufficient to distinguish between inflowing and outflowing gas. Additional properties, such as temperature, can help break this degeneracy. A velocity structure function (VSF) analysis indicates more coherent gas motion in the outskirts and more disturbed kinematics toward halo centers. In all cluster regions, the VSF shows a slope close to the theoretical models of Kolmogorov (∼1/3), except within 50 kpc of the cluster centers, where the slope is significantly steeper. The outcome of TNG-Cluster broadly aligns with observations of the VSF of multiphase gas across different scales in galaxy clusters, ranging from ∼1 kpc to megaparsec scales.
