Relay transmissions play an important role in many types of communication systems. In this paper, we consider packet-level coded transmissions over lossy relay links with the finite-buffer and coding coefficients delivery cost constraints. We propose a low-complexity coding scheme where packets are encoded from sequentially formed random subsets of source packets called batches. The relay recodes only from the buffered packets belonging to the same batch to maintain the code sparsity, lowering the packet header overhead and the decoding complexity compared to the random linear network coding (RLNC). To analyze the performance, we first propose an absorbing Markov chain model to analyze the RLNC transmission over finite-buffer relay links. The finite-length analysis not only provides a lower bound on the completion time using any sparser random codes but also characterizes each individual batch's transmission of the proposed code. Based on the analysis, another Markov chain is proposed to determine the decoding failure probability and the expected completion time of the batch coding scheme. As shown through analysis and simulations, the proposed scheme achieves higher effective end-to-end rates than RLNC when the coding coefficients delivery cost is considered and is also with much lowered decoding complexity thanks to its sparseness.