Integrating knowledge graphs (KGs) with large language models (LLMs) enhances factual accuracy and interpretability in question answering. However, existing agent-based methods rely on static memory mechanisms that fail to address the combinatorial explosion of search spaces in multi-hop reasoning and lack continuous learning capabilities. To overcome these limitations, we propose EvoMemKG, an agent framework with a dynamic, evolvable memory mechanism specifically designed for KG reasoning. EvoMemKG features a dual-layer memory architecture: (1) a working memory that losslessly compresses retrieved triplets through clustering to manage exploration states, effectively linearizing the exponential state space expansion; and (2) an experience memory that abstracts historical reasoning paths into reusable, generalized strategies, enabling cross-task knowledge transfer and self-evolution. We further introduce a double-loop workflow that orchestrates the LLM, memory layers, and KG environment to enable end-to-end autonomous reasoning. Extensive evaluations on three KGQA datasets across two KGs demonstrate that EvoMemKG achieves state-of-the-art performance without requiring additional training or specialized tools. Notably, it achieves improvements of up to 20% over the strong baseline on complex multi-hop queries, validating the effectiveness of our dynamic memory approach.