Edge-based, task-centric Federated Learning (FL) enabled by Mobile Edge Computing (MEC) and the Internet of Things (IoT) is essential to meeting the intelligent service demands of 6G networks. However, training task-specific FL models on resource-constrained devices remains a major challenge. This paper proposes a task-centric FL framework that leverages model transfer across distributed edge-cloud servers to enhance training efficiency and reduce overall latency. We model the end-to-end latency of FL, including model discovery, data routing, and training, and formulate a constrained optimization problem to minimize total latency through selective transfer of pre-trained FL models. Furthermore, we develop a maximum search budget-based solution that balances model search and transfer overhead with training performance. Simulation results demonstrate that the proposed approach significantly reduces FL latency compared to traditional solutions, providing a practical foundation for efficient FL over the edge-cloud continuum.