This study presents an Exploratory Retrieval-Augmented Planning (ExRAP) framework, designed to tackle continual instruction following tasks of embodied agents in dynamic, non-stationary environments. The framework enhances Large Language Models' (LLMs) embodied reasoning capabilities by efficiently exploring the physical environment and establishing the environmental context memory, thereby effectively grounding the task planning process in time-varying environment contexts. In ExRAP, given multiple continual instruction following tasks, each instruction is decomposed into queries on the environmental context memory and task executions conditioned on the query results. To efficiently handle these multiple tasks that are performed continuously and simultaneously, we implement an exploration-integrated task planning scheme by incorporating the information-based exploration into the LLM-based planning process. Combined with memory-augmented query evaluation, this integrated scheme not only allows for a better balance between the validity of the environmental context memory and the load of environment exploration, but also improves overall task performance. Furthermore, we devise a temporal consistency refinement scheme for query evaluation to address the inherent decay of knowledge in the memory. Through experiments with VirtualHome, ALFRED, and CARLA, our approach demonstrates robustness against a variety of embodied instruction following scenarios involving different instruction scales and types, and non-stationarity degrees, and it consistently outperforms other state-of-the-art LLM-based task planning approaches in terms of both goal success rate and execution efficiency.