Clinical Decision Support Systems (CDSS) have the potential to improve and standardise care with probabilistic guidance. However, many CDSS deploy static, generic rule-based logic, resulting in inequitably distributed accuracy and inconsistent performance in evolving clinical environments. Data-driven models could resolve this issue by updating predictions according to the data collected. However, the size of data required necessitates collaborative learning from analogous CDSS's, which are often imperfectly interoperable (IIO) or unshareable. We propose Modular Clinical Decision Support Networks (MoDN) which allow flexible, privacy-preserving learning across IIO datasets, as well as being robust to the systematic missingness common to CDSS-derived data, while providing interpretable, continuous predictive feedback to the clinician. MoDN is a novel decision tree composed of feature-specific neural network modules that can be combined in any number or combination to make any number or combination of diagnostic predictions, updatable at each step of a consultation. The model is validated on a real-world CDSS-derived dataset, comprising 3,192 paediatric outpatients in Tanzania. MoDN significantly outperforms 'monolithic' baseline models (which take all features at once at the end of a consultation) with a mean macro F1 score across all diagnoses of 0.749 vs 0.651 for logistic regression and 0.620 for multilayer perceptron (p < 0.001). To test collaborative learning between IIO datasets, we create subsets with various percentages of feature overlap and port a MoDN model trained on one subset to another. Even with only 60% common features, fine-tuning a MoDN model on the new dataset or just making a composite model with MoDN modules matched the ideal scenario of sharing data in a perfectly interoperable setting. MoDN integrates into consultation logic by providing interpretable continuous feedback on the predictive potential of each question in a CDSS questionnaire. The modular design allows it to compartmentalise training updates to specific features and collaboratively learn between IIO datasets without sharing any data.