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Motivated by the persistent anomalies reported in the $b\to cτ\barν$ data, we perform a general model-independent analysis of these transitions, in the presence of light right-handed neutrinos. We adopt an effective field theory approach and write a low-energy effective Hamiltonian, including all possible dimension-six operators. The corresponding Wilson coefficients are determined through a numerical fit to all available experimental data. In order to work with a manageable set of free parameters, we define eleven well-motivated scenarios, characterized by the different types of new physics that could mediate these transitions, and analyse which options seem to be preferred by the current measurements. The data exhibit a clear preference for new-physics contributions, and good fits to the data are obtained in several cases. However, the current measurement of the longitudinal $D^*$ polarization in $B\to D^*τ\barν$ cannot be easily accommodated within its experimental $1σ$ range. A general analysis of the three-body $B\to D τ\barν$ and four-body $B\to D^*(\to Dπ)τ\barν$ angular distributions is also presented. The accessible angular observables are studied in order to assess their sensitivity to the different new physics scenarios. Experimental information on these distributions would help to disentangle the dynamical origin of the current anomalies.