Flapping micro-air vehicles (MAVs) can access a wide range of locations, including confined spaces such as the inside of industrial plants and collapsed buildings, and offer high maneuverability and tolerance to disturbances. However, current flapping MAVs require transmission systems between their actuators and wings, which introduce energetic losses and additional mass, hindering performance. Here, we introduce a high-performance electrostatic flapping actuation system, the liquid-amplified zipping actuator (LAZA), which induces wing movement by direct application of liquid-amplified electrostatic forces at the wing root, eliminating the requirement of any transmission system and their associated downsides. The LAZA allows for accurate control of flapping frequency and amplitude, exhibits no variation in performance over more than 1 million actuation cycles, and delivers peak and average specific powers of 200 and 124 watts per kilogram, respectively, exceeding mammalian and insect flight muscle and on par with modern flapping MAV actuation systems. The inclusion of 50-millimeter-long passively pitching wings in a dragonfly-sized LAZA flapping system allowed the rectification of net directional thrust up to 5.73 millinewtons. This thrust was achieved while consuming only 243 milliwatts of electrical power, implying a thrust-to-power ratio of 23.6 newtons per kilowatt, similar to state-of-the-art flapping MAVs, helicopter rotors, and commercial drone motors. Last, a horizontally moving LAZA flapping system supported by a taut nylon wire was able to accelerate from at-rest and travel at speeds up to 0.71 meters per second. The LAZA enables lightweight, high-performance transmission-free flapping MAVs for long-term remote exploration and search-and-rescue missions.