We study and analyze the performance of a wireless communication system that is based on the use of airborne Reconfigurable Intelligent Surfaces, i.e., surfaces mounted on an Unmanned Aerial Vehicle (UAV). To this end, we perform a stochastic analysis that allows us to study a very wide variety of realistic channel conditions, including environments characterized by combined small-scale and large-scale fading and potential spatial correlation in shadowing. For this generic channel model, we derive closed-form expressions for the outage probability and the ergodic capacity, and also investigate the analytical calculation of the average energy efficiency. The presented results allow us, for the first time, to study the impact of different shadowing conditions, including correlation effects, in aerial communications systems that are also supported by reconfigurable intelligent surfaces. Finally, the performance of the considered system is also compared with that of decodeand-forward relaying, highlighting the advantages of combining reconfigurable intelligent surfaces with UAV-assisted communication technologies in composite fading environments.