Further accelerated by the Internet of Things (IoT) concept, various devices are being continuously introduced into diverse application scenarios. To achieve unattended updates of IoT smart object(s), there remains a challenging problem concerning how to efficiently deliver messages to specific groups of target nodes, especially considering node mobility. In this paper, the relay selection problem is investigated on the basis of directional movement with randomness (e.g. typically associated with the searching or migrating behaviour of animals). Unlike numerous works tackling one-to-one communication, we focus on efficient group delivery (one-to-many). A two-level delay budget model is considered to reflect the flexibility of delay tolerance, which brings potential efficiency gains for group delivery compared with using a single budget boundary. Following the description of the system model, a combinatorial bi-objective optimisation problem is formulated and solutions are proposed. Simulation results show that the greedy algorithm can achieve comparable performance to an evolutionary algorithm when the delivery satisfaction outweighs efficiency. Furthermore, we show that our proposed greedy scheme can outperform the state-of-the-art when the delivery efficiency becomes increasingly important.