Federated domain-specific instruction tuning (FedDIT) for large language models (LLMs) aims to enhance performance in specialized domains using distributed private and limited data, yet identifying key performance drivers and optimal augmentation strategies remains challenging. We empirically establish that cross-client domain coverage, rather than data heterogeneity, is the pivotal factor. We then introduce FedDCA, an algorithm that explicitly maximizes this coverage through diversity-oriented client center selection and retrieval-based augmentation, constructing diverse, non-redundant cross-client instruction sets. Extensive experiments across multiple domains demonstrate FedDCA's superiority over eleven baselines, achieving performance gains of up to 29.19\% and domain coverage improvements of 4.82\%-21.36\%. FedDCA maintains its effectiveness in diverse and challenging scenarios, including data selection, held-out settings where task-specific public data is scarce and various data heterogeneity, with manageable privacy risks. This work clarifies critical FedDIT dynamics and presents FedDCA as an effective, privacy-preserving, and scalable solution for advancing domain-specific LLM tuning.