Due to the distributed characteristics of Federated Learning (FL), the vulnerability of global model and coordination of devices are the main obstacle. As a promising solution of decentralization, scalability and security, leveraging blockchain in FL has attracted much attention in recent years. However, the traditional consensus mechanisms designed for blockchain like Proof of Work (PoW) would cause extreme resource consumption, which reduces the efficiency of FL greatly, especially when the participating devices are wireless and resource-limited. In order to address device asynchrony and anomaly detection in FL while avoiding the extra resource consumption caused by blockchain, this paper introduces a framework for empowering FL using Direct Acyclic Graph (DAG)-based blockchain systematically (DAG-FL). Accordingly, DAG-FL is first introduced from a three-layer architecture in details, and then two algorithms DAG-FL Controlling and DAG-FL Updating are designed running on different nodes to elaborate the operation of DAG-FL consensus mechanism. After that, a Poisson process model is formulated to discuss that how to set deployment parameters to maintain DAG-FL stably in different federated learning tasks. The extensive simulations and experiments show that DAG-FL can achieve better performance in terms of training efficiency and model accuracy compared with the typical existing on-device federated learning systems as the benchmarks.