Federated Learning (FL) provides a privacy-preserving framework for distributed machine learning but struggles with challenges arising from client heterogeneity and non-iid data distributions. These challenges often result in slower model convergence, reduced training efficiency, and compromised fairness and accuracy, especially when clients with unique data are excluded from training rounds. To overcome these issues, we introduce CrossFL, an innovative FL framework designed to ensure meaningful contributions from all clients, including those omitted in earlier rounds. CrossFL incorporates a cross-round training mechanism with inclusive aggregation, enabling the integration of model updates from previously unselected clients, thereby enhancing global model diversity and improving convergence in non-iid scenarios. Furthermore, a grouped asynchronous client selection strategy eliminates synchronization delays by processing updates as they are received, reducing training time and improving adaptability in heterogeneous environments. Experimental evaluations reveal that CrossFL achieves at least 1.42 \(\times \) faster convergence and improves accuracy by 0.68% to 12.06% compared to existing methods. It effectively handles extreme data distributions and is compatible with various optimization algorithms. By addressing critical limitations in client selection and aggregation strategies, CrossFL establishes a scalable and efficient approach for FL across diverse and real-world scenarios.