This paper discusses the future of migrants with acute heart problems and without permanent permission to remain in the country where they are seeking asylum. What does the country they have traveled to owe them? Specifically, what healthcare services are they entitled to? This may seem a niche problem, but numbers of migrants with acute heart problems could increase in the future. Besides, similar problems could be raised by, for instance, traumatized migrants with acute needs for healthcare services for other serious conditions. The paper identifies the issues and some positions on them. Arguments for and against these positions are explored. This particular set of problems in healthcare ethics creates several challenges, at both national and international levels, concerning access to transplantation, public willingness to donate organs, optimal use of organs, justice and fairness, and potential conflicts of law, politics and ethics, as well as issues revolving around interaction and communication (or lack of it) between agencies and professions.