This article focuses on designing high-data-rate wireless communications for drone networks in the mmWave and terahertz (THz) frequency bands. MmWave/THz-band communications have been envisioned as key technologies to achieve ultra broadband wireless links through beamforming in 5G and beyond networks. However, a main challenge with these frequency bands is that the narrow-beam directional wireless links can be easily disconnected because of the beam misalignment in mobile environments. To address this challenge, in this article we design a new beam control scheme called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LeBeam</i> , with the objective of maximizing the expected capacity of the mmWave/THz-band links by determining the optimal beamwidth dynamically under the mobility uncertainties of flying drones. In <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LeBeam</i> , an Echo State Network (ESN) is adopted to capture the mobility uncertainties of the drones dynamically and predict the optimal beamwidth based on the first- and second-order moments of the drone mobility. The ESN has been trained based on real drone flight traces. To this end, we measure and analyze the mobility uncertainties of flying drones by carrying out a series of field experiments in different weather. It is found that flying drones experience micro-, small- and large-scale mobility uncertainties, and the resulting mobility behavior cannot be characterized with any existing statistical models. The performance of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LeBeam</i> is evaluated over UBSim, a newly developed trace-driven Universal Broadband Simulator for integrated aerial and ground wireless networking. Results indicate that the micro-scale mobility has only negligible effects on the link capacity (less than 1 percent), while the wireless links may experience significant capacity degradation (over 50 percent on average) in the presence of small- and large-scale mobility uncertainties.