Recent availability of geo-localized data capturing individual human activity together with the statistical data on international migration opened up unprecedented opportunities for a study on global mobility. In this paper, we consider it from the perspective of a multi-layer complex network, built using a combination of three datasets: Twitter, Flickr and official migration data. Those datasets provide different, but equally important insights on the global mobility - while the first two highlight short-term visits of people from one country to another, the last one - migration - shows the long-term mobility perspective, when people relocate for good. The main purpose of the paper is to emphasize importance of this multi-layer approach capturing both aspects of human mobility at the same time. On the one hand, we show that although the general properties of different layers of the global mobility network are similar, there are important quantitative differences among them. On the other hand, we demonstrate that consideration of mobility from a multi-layer perspective can reveal important global spatial patterns in a way more consistent with those observed in other available relevant sources of international connections, in comparison to the spatial structure inferred from each network layer taken separately.