Mobile Low-Duty-Cycle Wireless Sensor Network (MLDC-WSN) is a new type of wireless multi-hop network composed of sensor nodes that have mobility and sleep for a long time in a wireless self-organizing manner. Neighbor discovery protocols are used for sensor nodes to discover neighbor nodes within their communication range and can provide neighbor information to other protocols. MLDC-WSN uses low duty cycle technology to reduce energy consumption, and captures dynamic environmental information through the movement of nodes. However, the random mobility and long dormancy of nodes lead to frequent changes in the network topology, and nodes need to consume a lot of energy and time to discover all their neighbors. The sensor nodes are often powered by energy-limited batteries, and battery replacement is difficult. How to implement efficient neighbor discovery with low-latency and low-energy is a challenge of current research. Different from the previous slot model that couples the listening and transmitting of beacons in one slot, this paper proposes a new discovery schedule based on the slot model that separates the listening and transmitting of beacons. The discovery schedule can guarantee unidirectional discovery. And the node proactively transmits the beacon to accelerate bidirectional discovery after unilaterally discovering its neighbor nodes. On this basis, a proactive asynchronous neighbor discovery protocol called Fedab is proposed in this paper. Ignoring beacon collisions and other realistic interference factors, Fedab can enable neighbor discovery within the theoretical worst-case discovery latency. Moreover, Fedab is extended to the scenario where nodes perform different duty cycles, i.e., asymmetric scenarios. After experimental simulations and numerical analysis, the Fedab protocol has better performance compared to other existing neighbor discovery protocols.