In future wireless networks, we anticipate that a large number of devices will connect to mobile networks through moving relays installed on vehicles, in particular in public transport vehicles. To provide high-speed moving relays with accurate channel state information different methods have been proposed, among which predictor antenna (PA) is one of the promising ones. Here, the PA system refers to a setup where two sets of antennas are deployed on top of a vehicle, and the front antenna(s) can be used to predict the channel state information for the antenna(s) behind. In this paper, we study the delay-limited performance of PA systems using adaptive rate allocations. We use the fundamental results on the achievable rate of finite block-length codes to study the system throughput and error probability in the presence of short packets. Particularly, we derive closed-form expressions for the error probability, the average transmit rate as well as the optimal rate allocation, and study the effect of different parameters on the performance of PA systems. The results indicate that rate adaptation under finite block-length codewords can impro