Supervised learning for audio classification typically imposes a fixed class vocabulary, which can be limiting for real-world applications where the target class vocabulary is not known a priori or changes dynamically. In this work, we introduce a few-shot continual learning framework for audio classification, where we can continuously expand a trained base classifier to recognize novel classes based on only few labeled data at inference time. This enables fast and interactive model updates by end-users with minimal human effort. To do so, we leverage the dynamic few-shot learning technique and adapt it to a challenging multi-label audio classification scenario. We incorporate a recent state-of-the-art audio feature extraction model as a backbone and perform a comparative analysis of our approach on two popular audio datasets (ESC-50 and AudioSet). We conduct an in-depth evaluation to illustrate the complexities of the problem and show that, while there is still room for improvement, our method outperforms three baselines on novel class detection while maintaining its performance on base classes.