We propose a novel approach to building influence-driven explanations (IDXs) for (discrete) Bayesian network classifiers (BCs). IDXs feature two main advantages wrt other commonly adopted explanation methods. First, IDXs may be generated using the (causal) influences between intermediate, in addition to merely input and output, variables within BCs, thus providing a deep, rather than shallow, account of the BCs’ behaviour. Second, IDXs are generated according to a configurable set of properties, specifying which influences between variables count towards explanations. Our approach is thus flexible and can be tailored to the requirements of particular contexts or users. Leveraging on this flexibility, we propose novel IDX instances as well as IDX instances capturing existing approaches. We demonstrate IDXs’ capability to explain various forms of BCs, and assess the advantages of our proposed IDX instances with both theoretical and empirical analyses.