We explore the use of orbital-angular-momentum (OAM)-multiplexing to increase the capacity of free-space data transmission to moving platforms, with an added potential benefit of decreasing the probability of data intercept. Specifically, we experimentally demonstrate and characterize the performance of an OAM-multiplexed, free-space optical (FSO) communications link between a ground transmitter and a ground receiver via a moving unmanned-aerial-vehicle (UAV). We achieve a total capacity of 80 Gbit/s up to 100-m-roundtrip link by multiplexing 2 OAM beams, each carrying a 40-Gbit/s quadrature-phase-shift-keying (QPSK) signal. Moreover, we investigate for static, hovering, and moving conditions the effects of channel impairments, including: misalignments, propeller-induced airflows, power loss, intermodal crosstalk, and system bit error rate (BER). We find the following: (a) when the UAV hovers in the air, the power on the desired mode fluctuates by 2.1 dB, while the crosstalk to the other mode is -19 dB below the power on the desired mode; and (b) when the UAV moves in the air, the power fluctuation on the desired mode increases to 4.3 dB and the crosstalk to the other mode increases to -10 dB. Furthermore, the channel crosstalk decreases with an increase in OAM mode spacing.