Teleoperation is a key paradigm for transferring human dexterity to robots, yet most prior work targets objects that are initially static, such as grasping or manipulation. Dynamic object catch, where objects move before contact, remains underexplored. Pure teleoperation in this task often fails due to timing, pose, and force errors, highlighting the need for shared autonomy that combines human input with autonomous policies. To this end, we present Tele-Catch, a systematic framework for dexterous hand teleoperation in dynamic object catching. At its core, we design DAIM, a dynamics-aware adaptive integration mechanism that realizes shared autonomy by fusing glove-based teleoperation signals into the diffusion policy denoising process. It adaptively modulates control based on the interaction object state. To improve policy robustness, we introduce DP-U3R, which integrates unsupervised geometric representations from point cloud observations into diffusion policy learning, enabling geometry-aware decision making. Extensive experiments demonstrate that Tele-Catch significantly improves accuracy and robustness in dynamic catching tasks, while also exhibiting consistent gains across distinct dexterous hand embodiments and previously unseen object categories.