Global health as a discipline is at a moment of tremendous achievements and general optimism, but faces enduring questions, notably about the sustainability of its endeavours. The field of global health defines itself through enumerated goals, targets, and indicators to track progress of complex changes. With this comes a broad panoply of objective-driven strategies resting on assumed control over chains of cause and effect. We present areas where emergence and self-organization play an important role in determining the sustainability of positive outcomes at scale and behaviours of both the population and its service providers. The role of self-organization and emergence seem under-appreciated by the field and we consider three reasons for a possible "blind spot" of global health decision makers and influencers: (1) the culture of command and control inherent to central planning, (2) the structural limitations of large complex institutional edifices in processing information (variety), and (3) epistemological/cognitive constraints. More pro-active allowance for the conditions of positive self-organization and emergence may be required to avoid unintended and negative long-term consequences of global strategies, in spite of best intentions. Observational, analytical and computational multi-disciplinary studies are needed to bring clarity to these questions, and to inform a parallel process of dialogue, provocation, and learning to evolve the institutional culture of global health at both national and international levels.