If you babysit and tutor younger children after school, the kids are under your tutelage. You are responsible for their care and education.
Tutelage can mean guardianship as well as teaching and sometimes it's difficult to tell which sense is meant. If an athlete is under the tutelage of his coach, the coach teaches him but is also responsible for the athlete's health and well-being. When the word describes a situation where one country or culture takes charge of another, it's more clear that management and guardianship are meant.