In child-robot interaction (cHRI) research, many studies pursue the goal to develop interactive systems that can be applied in everyday settings. For early education, increasingly, the setting of a kindergarten is targeted. However, when cHRI and research are brought into a kindergarten, a range of ethical and related procedural aspects have to be considered and dealt with. While ethical models elaborated within other human-robot interaction settings, e.g., assisted living contexts, can provide some important indicators for relevant issues, we argue that it is important to start developing a systematic approach to identify and tackle those ethical issues which rise with cHRI in kindergarten settings on a more global level and address the impact of the technology from a macroperspective beyond the effects on the individual. Based on our experience in conducting studies with children in general and pedagogical considerations on the role of the institution of kindergarten in specific, in this paper, we enfold some relevant aspects that have barely been addressed in an explicit way in current cHRI research. Four areas are analyzed and key ethical issues are identified in each area: (1) the institutional setting of a kindergarten, (2) children as a vulnerable group, (3) the caregivers' role, and (4) pedagogical concepts. With our considerations, we aim at (i) broadening the methodology of the current studies within the area of cHRI, (ii) revalidate it based on our comprehensive empirical experience with research in kindergarten settings, both laboratory and real-world contexts, and (iii) provide a framework for the development of a more systematic approach to address the ethical issues in cHRI research within kindergarten settings.