We study a stochastic multi-armed bandit problem in which the set of available arms expands over time. This setting arises in sequential experimentation when new actions or treatments become available during an ongoing study, making regret against a single best arm in hindsight inappropriate. We instead evaluate performance relative to the best arm currently available, leading to a dynamic-regret criterion for arriving-arm environments. To address the resulting challenges of arrival information discrepancy (AID) and a drifting benchmark (DB), we propose UCB for Arriving Arms (UCB-AA), an elimination-based procedure with an aiding preliminary screening step for newly arrived arms before full competition with incumbent arms. We show that UCB-AA attains regret bounds that depend explicitly on the arrival process, achieves sublinear dynamic regret under regularity conditions on gap evolution, and admits an online extension for unknown horizons. Simulation results show that UCB-AA reduces wasted pulls and maintains a smaller active arm set while preserving competitive regret performance.