The article deals with one of the effects of health inequalities and gaps in access to treatments for rare diseases, namely health-driven emigration. The purpose of the paper is to systematize knowledge about the phenomenon of health emigration observed among families affected by rare diseases, for which reimbursed treatment is available, but only in selected countries. The topic proved to be niche; the issue of “health emigration in rare diseases” is an area for exploration. Therefore, the further analysis used text mining and machine learning methods based on a database selected based on keywords related to this issue. The results made it possible to systematize the guesses made by researchers in management and economic fields, to identify the most common keywords and thematic clusters around the perspective of the patient, drug manufacturer and treatment reimbursement decision-maker, and the perspective integrating all the others. Since the topic of health emigration was not directly addressed in the selected sources, the authors attempted to define the related concepts and discussed the importance of this phenomenon in managing the support system in rare diseases. Thus, they indicated directions for further research in this area.