Cross-domain reinforcement learning (RL) aims to learn transferable policies under dynamics shifts between source and target domains. A key challenge lies in the lack of target-domain environment interaction and reward supervision, which prevents direct policy learning. To address this challenge, we propose Bridging Dynamics Gaps for Cross-Domain Reinforcement Learning (BDGxRL), a novel framework that leverages Diffusion Schr\"odinger Bridge (DSB) to align source transitions with target-domain dynamics encoded in offline demonstrations. Moreover, we introduce a reward modulation mechanism that estimates rewards based on state transitions, applying to DSB-aligned samples to ensure consistency between rewards and target-domain dynamics. BDGxRL performs target-oriented policy learning entirely within the source domain, without access to the target environment or its rewards. Experiments on MuJoCo cross-domain benchmarks demonstrate that BDGxRL outperforms state-of-the-art baselines and shows strong adaptability under transition dynamics shifts.