Deep Learning-based end-to-end Automatic Speech Recognition (ASR) has made significant strides but still struggles with performance on out-of-domain samples due to domain shifts in real-world scenarios. Test-Time Adaptation (TTA) methods address this issue by adapting models using test samples at inference time. However, current ASR TTA methods have largely focused on non-continual TTA, which limits cross-sample knowledge learning compared to continual TTA. In this work, we first propose a Fast-slow TTA framework for ASR that leverages the advantage of continual and non-continual TTA. Following this framework, we introduce Dynamic SUTA (DSUTA), an entropy-minimization-based continual TTA method for ASR. To enhance DSUTA robustness for time-varying data, we design a dynamic reset strategy to automatically detect domain shifts and reset the model, making it more effective at handling multi-domain data. Our method demonstrates superior performance on various noisy ASR datasets, outperforming both non-continual and continual TTA baselines while maintaining robustness to domain changes without requiring domain boundary information.