In this paper, we consider a hybrid satellite-terrestrial network (HSTN) where a multiantenna satellite communicates with a ground user equipment (UE) with the help of multiple cache-enabled amplify-and-forward (AF) three-dimensional ($3$D) mobile unmanned aerial vehicle (UAV) relays. Herein, we employ the two fundamental most popular content (MPC) and uniform content (UC) caching schemes for two types of mobile UAV relays, namely fully $3$D and fixed height. Taking into account the multiantenna satellite links and the random $3$D distances between UAV relays and UE, we analyze the outage probability (OP) of considered system with MPC and UC caching schemes. We further carry out the corresponding asymptotic OP analysis to present the insights on achievable performance gains of two schemes for both types of $3$D mobile UAV relaying. Specifically, we show the following: (a) MPC caching dominates the UC and no caching schemes; (b) fully $3$D mobile UAV relaying outperforms its fixed height counterpart. We finally corroborate the theoretic analysis by simulations.