This is the second paper of a series exploring the multi-component (stars, warm and cold gas and radio jets) properties of a sample of eleven nearby low excitation radio galaxies (LERGs), with the aim of better understanding the AGN fuelling/feedback cycle in these objects. Here we present a study of the molecular gas kinematics of six sample galaxies detected in $^{12}$CO(2-1) with ALMA. In all cases, our modelling suggests that the bulk of the gas in the observed (sub-)kpc CO discs is in ordered rotation. Nevertheless, low-level distortions are ubiquitous, indicating that the molecular gas is not fully relaxed into the host galaxy potential. The majority of the discs, however, are only marginally resolved, preventing us from drawing strong conclusions. NGC 3557 and NGC 3100 are special cases. The features observed in the CO velocity curve of NGC 3557 allow us to estimate a super-massive black hole (SMBH) mass of $(7.10pm0.02)times10^{8}$ M$_{odot}$, in agreement with expectations from the M$_{rm SMBH}- sigma_{*}$ relation. The rotation pattern of NGC 3100 shows distortions that appear to be consistent with the presence of both a position angle and inclination warp. Non-negligible radial motions are also found in the plane of the CO disc, likely consistent with streaming motions associated with the spiral pattern found in the inner regions of the disc. The dominant radial motions are likely to be inflows, supporting a scenario in which the cold gas is contributing to the fuelling of the AGN.
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