Recent speech foundation models excel at multilingual automatic speech recognition (ASR) for high-resource languages, but adapting them to low-resource languages remains challenging due to data scarcity and efficiency constraints. Full-model fine-tuning is computationally expensive and prone to overfitting, while parameter-efficient methods like LoRA apply adaptation uniformly across layers, overlooking internal representations thus compromising effectiveness and efficiency. We analyze multilingual ASR models and reveal a U-shaped adaptability pattern: early and late layers are language-specific and require more adaptation, while intermediate layers retain shared semantics and need less. Building on this observation, we propose DAMA, a Depth-Aware Model Adaptation framework that allocates adaptation capacity according to each layer's role. DAMA also introduces Singular Value Decomposition (SVD)-based initialization to constrain adaptation and preserve the U-shaped pattern, as well as a frozen middle-layer basis for further efficiency. Evaluated on 18 low-resource languages across two benchmark datasets, DAMA matches or surpasses state-of-the-art accuracy with 80% fewer trainable parameters, achieves a 29% error reduction under extreme data scarcity, and significantly improves memory, training time, and computational efficiency over baselines. These results highlight the benefits of structure-aware adaptation for efficient, scalable multilingual ASR.