Recent progress in Neural Causal Models (NCMs) showcased how identification and partial identification of causal effects can be automatically carried out via training of neural generative models that respect the constraints encoded in a given causal graph [Xia et al. 2022, Balazadeh et al. 2022]. However, formal consistency of these methods has only been proven for the case of discrete variables or only for linear causal models. In this work, we prove consistency of partial identification via NCMs in a general setting with both continuous and categorical variables. Further, our results highlight the impact of the design of the underlying neural network architecture in terms of depth and connectivity as well as the importance of applying Lipschitz regularization in the training phase. In particular, we provide a counterexample showing that without Lipschitz regularization the NCM may not be asymptotically consistent. Our results are enabled by new results on the approximability of structural causal models via neural generative models, together with an analysis of the sample complexity of the resulting architectures and how that translates into an error in the constrained optimization problem that defines the partial identification bounds.