Learning-based approaches to monocular motion capture have recently shown promising results by learning to regress in a data-driven manner. However, due to the challenges in data collection and network designs, it remains challenging for existing solutions to achieve real-time full-body capture while being accurate in world space. In this work, we introduce ProxyCap, a human-centric proxy-to-motion learning scheme to learn world-space motions from a proxy dataset of 2D skeleton sequences and 3D rotational motions. Such proxy data enables us to build a learning-based network with accurate world-space supervision while also mitigating the generalization issues. For more accurate and physically plausible predictions in world space, our network is designed to learn human motions from a human-centric perspective, which enables the understanding of the same motion captured with different camera trajectories. Moreover, a contact-aware neural motion descent module is proposed in our network so that it can be aware of foot-ground contact and motion misalignment with the proxy observations. With the proposed learning-based solution, we demonstrate the first real-time monocular full-body capture system with plausible foot-ground contact in world space even using hand-held moving cameras.