Temporal Entity Alignment (TEA) aims to identify equivalent entities across Temporal Knowledge Graphs (TKGs). Existing TEA methods generally assume static snapshots and repeatedly perform full-graph alignment, resulting in high computational costs. Recent evolving EA models handle continuously changing graphs through incremental alignment strategies, but mainly focus on non-temporal KGs and overlook temporal dynamics. Consequently, Evolving TEA, a realistic scenario where both graph structures and temporal information evolve over time, remains underexplored. To bridge this gap, we propose a lightweight continual TEA framework with three novel design: (1) informative seed selection that jointly leverages structural and temporal signals for reliable iterative finetuning, (2) localized inference that restricts computation to newly updated entities and their neighborhoods for efficient alignment, and (3) cross-snapshot score fusion that adaptively integrates historical and current alignment evidence. Extensive experiments on real-world evolving TEA benchmarks demonstrate the superiority of our proposed framework, achieving state-of-the-art alignment performance while significantly reducing computational overhead.