Satellite and unmanned aerial vehicle (UAV) integrated networks (SUINs) are considered as a promising method to offer various Internet of Remote Things (IoRT) applications. In this article, we investigate the downlink transmission of SUINs where the satellite-to-UAV link uses the free-space optical (FSO) technology with an equal gain combining (EGC) scheme while the links from UAV to IoRT devices exploit radio frequency (RF) with the space-division multiple access (SDMA) technique. Specifically, considering that only statistical channel state information (CSI) is available, we first formulate an optimization problem to maximize the ergodic sum rate (ESR) of the system, which is constrained by the total transmit power budget and IoRT devices’ rate requirements. Then, a beamforming (BF) scheme based on the alternating direction method of multipliers (ADMM) is proposed to solve the nonconvex problem. Furthermore, a zero-forcing (ZF)-based suboptimal approach is also presented to reduce the implementation complexity. Finally, by assuming that the FSO link and RF links are subject to Gamma–Gamma fading and Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> fading, respectively, we derive closed-form ESR expressions for the considered network with the proposed BF schemes. Simulation results are provided to confirm the accuracy of the theoretical analysis. Moreover, it is revealed that our proposed EGC scheme for FSO communication and BF schemes for RF transmission can both achieve better performance than the existing works.