In recent years, cross-national collaboration in medical research has gained increased policy attention. Policies are developed to enhance data sharing, ensure open-access, and harmonize international standards and ethics rules in order to promote access to existing resources and increase scientific output. In tandem with this promotion of data sharing, numerous ethics policies are developed to control data flows and protect privacy and confidentiality. Both sets of policy making, however, pay limited attention to the moral decisions and social ties enacted in the everyday routines of scientific work. This paper takes its point of departure in the practices of a Danish laboratory with great experience in international collaboration regarding genetic research. We focus on a simple query, what makes genetic material and health data flow, and which hopes and concerns travel along with them? We explore what we call the flows, the nonflows, and the overflows of material and information, and we document the work producing the flows of health data and biomaterial. We call this work "ethics work" and argue that it is crucial for data sharing though it is rarely articulated in ethics policies, remains inadequately funded, and lacks acknowledgment in policies promoting international data sharing.