In today's landscape, Mixture of Experts (MoE) is a crucial architecture that has been used by many of the most advanced models. One of the major challenges of MoE models is that they usually require much more memory than their dense counterparts due to their unique architecture, and hence are harder to deploy in environments with limited GPU memory, such as edge devices. MoE offloading is a promising technique proposed to overcome this challenge, especially if it is enhanced with caching and pre-fetching, but prior work stopped at suboptimal caching algorithm and offered limited insights. In this work, we study MoE offloading in depth and make the following contributions: 1. We analyze the expert activation and LRU caching behavior in detail and provide traces. 2. We propose LFU caching optimization based on our analysis and obtain strong improvements from LRU. 3. We implement and experiment speculative expert pre-fetching, providing detailed trace showing its huge potential . 4. In addition, our study extensively covers the behavior of the MoE architecture itself, offering information on the characteristic of the gating network and experts. This can inspire future work on the interpretation of MoE models and the development of pruning techniques for MoE architecture with minimal performance loss.