AI-driven respiratory sound classification (RSC) is promising for automated pulmonary disease detection, yet multi-site deployment is hindered by inter-stethoscope variability. We introduce a federated domain generalization (FedDG) formulation for RSC under stethoscope-induced device shifts, where clients use heterogeneous devices and the model is evaluated on unseen devices. Our empirical analysis shows that stethoscope-induced style and disease-specific content are tightly entangled, making deterministic style removal unreliable. In response, we propose a causality-inspired multimodal FedDG framework that combines: (i) a causality-inspired device style intervention network that performs content-preserving style perturbations, (ii) counterfactual text augmentation that neutralizes metadata shortcuts, and (iii) gradient alignment that facilitates device-invariant representations across clients. Built on a multimodal language-audio pretraining model, it outperforms conventional data augmentation and federated learning baselines in leave-one-device-out validation on ICBHI and SPRSound datasets. Code will be released upon publication.