Class imbalance introduces additional challenges when learning classifiers from concept drifting data streams. Most existing work focuses on designing new algorithms for dealing with the global imbalance ratio and does not consider other data complexities. Independent research on static imbalanced data has highlighted the influential role of local data difficulty factors such as minority class decomposition and presence of unsafe types of examples. Despite often being present in real-world data, the interactions between concept drifts and local data difficulty factors have not been investigated in concept drifting data streams yet. We thoroughly study the impact of such interactions on drifting imbalanced streams. For this purpose, we put forward a new categorization of concept drifts for class imbalanced problems. Through comprehensive experiments with synthetic and real data streams, we study the influence of concept drifts, global class imbalance, local data difficulty factors, and their combinations, on predictions of representative online classifiers. Experimental results reveal the high influence of new considered factors and their local drifts, as well as differences in existing classifiers’ reactions to such factors. Combinations of multiple factors are the most challenging for classifiers. Although existing classifiers are partially capable of coping with global class imbalance, new approaches are needed to address challenges posed by imbalanced data streams.