Federated learning (FL) has emerged as a key enabler of distributed model training across ubiquitous wireless edge devices while preserving data privacy in next-generation wireless systems. One communication-efficient approach to realize this is leveraging Over-the-Air Federated Learning (OTA-FL), which exploits the superposition property of wireless multiple access channels (MAC) to perform low-latency model aggregation. However, practical OTA-FL deployments are significantly affected by the joint effects of channel noise at the server, fading, and both data and system heterogeneity across participating edge devices, which can degrade the model performance and thereby convergence. In this work, we propose the novel Robust and Convergent Federated Learning (RoCo-FL), a framework that jointly handles these challenges in federated wireless networks. RoCo-FL formulates local updates through inexact optimization, capturing the effects of wireless noise, partial participation, and heterogeneous system capabilities via structured inexactness conditions. These conditions naturally manifest as a proximal constraint at the edge devices that stabilizes local updates and mitigates the impact of channel impairments, data heterogeneity, and stragglers. We establish convergence guarantees for non-convex objectives under the proposed zeroth- and first-order inexactness models, providing theoretical insight into the interplay between channel impairments, data, and system heterogeneity. Experimentally, we validate RoCo-FL on real-world datasets, including CIFAR10, CIFAR100, and FEMNIST, demonstrating its robustness in noisy and heterogeneous environments. Compared to state-of-the-art baselines, RoCo-FL achieves significantly more stable convergence and higher accuracy, particularly in the presence of stragglers.

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