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This paper introduces MobileH2R, a framework for learning generalizable vision-based human-to-mobile-robot (H2MR) handover skills. Unlike traditional fixed-base handovers, this task requires a mobile robot to reliably receive objects in a large workspace enabled by its mobility. Our key insight is that generalizable handover skills can be developed in simulators using high-quality synthetic data, without the need for real-world demonstrations. To achieve this, we propose a scalable pipeline for generating diverse synthetic full-body human motion data, an automated method for creating safe and imitation-friendly demonstrations, and an efficient 4D imitation learning method for distilling large-scale demonstrations into closed-loop policies with base-arm coordination. Experimental evaluations in both simulators and the real world show significant improvements (at least +15% success rate) over baseline methods in all cases. Experiments also validate that large-scale and diverse synthetic data greatly enhances robot learning, highlighting our scalable framework.