The Reverse $k$-Nearest Neighbor (R$k$NN) query over moving objects on road networks seeks to find all moving objects that consider the specified query point as one of their $k$ nearest neighbors. In location based services, many users probably submit R$k$NN queries simultaneously. However, existing methods largely overlook how to efficiently process multiple such queries together, missing opportunities to share redundant computations and thus reduce overall processing costs. To address this, this work is the first to explore batch processing of multiple R$k$NN queries, aiming to minimize total computation by sharing duplicate calculations across queries. To tackle this issue, we propose the BR$k$NN-Light algorithm, which uses rapid verification and pruning strategies based on geometric constraints, along with an optimized range search technique, to speed up the process of identifying the R$k$NNs for each query. Furthermore, it proposes a dynamic distance caching mechanism to enable computation reuse when handling multiple queries, thereby significantly reducing unnecessary computations. Experiments on multiple real-world road networks demonstrate the superiority of the BR$k$NN-Light algorithm on the processing of batch queries.