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Stereo Depth Estimation in real-world environments poses significant challenges due to dynamic domain shifts, sparse or unreliable supervision, and the high cost of acquiring dense ground-truth labels. While recent Test-Time Adaptation (TTA) methods offer promising solutions, most rely on static target domain assumptions and input-invariant adaptation strategies, limiting their effectiveness under continual shifts. In this paper, we propose RobIA, a novel Robust, Instance-Aware framework for Continual Test-Time Adaptation (CTTA) in stereo depth estimation. RobIA integrates two key components: (1) Attend-and-Excite Mixture-of-Experts (AttEx-MoE), a parameter-efficient module that dynamically routes input to frozen experts via lightweight self-attention mechanism tailored to epipolar geometry, and (2) Robust AdaptBN Teacher, a PEFT-based teacher model that provides dense pseudo-supervision by complementing sparse handcrafted labels. This strategy enables input-specific flexibility, broad supervision coverage, improving generalization under domain shift. Extensive experiments demonstrate that RobIA achieves superior adaptation performance across dynamic target domains while maintaining computational efficiency.