We investigate pneumatic non-prehensile manipulation (i.e., blowing) as a means of efficiently moving scattered objects into a target receptacle. Due to the chaotic nature of aerodynamic forces, a blowing controller must (i) continually adapt to unexpected changes from its actions, (ii) maintain fine-grained control, since the slightest misstep can result in large unintended consequences (e.g., scatter objects already in a pile), and (iii) infer long-range plans (e.g., move the robot to strategic blowing locations). We tackle these challenges in the context of deep reinforcement learning, introducing a hierarchical version of the spatial action maps framework. This allows for efficient learning of multi-level vision-based policies that effectively combine high-level planning and low-level closed-loop control for dynamic mobile manipulation. Experiments show that our system learns efficient behaviors for the task, demonstrating in particular that blowing achieves better downstream performance than pushing, and that our hierarchical policies improve performance over flat baselines. Moreover, we show that our method naturally encourages emergent specialization between levels of the hierarchy spanning low-level fine-grained control and high-level planning. We also demonstrate qualitative results on a real mobile robot equipped with a miniature air blower.