I don't think that there is a lot of adult directed passing on to children of singing games among Americans of which ever race/ethnicity. These games are usually thought to be too "babyish" for children over pre-school ages (age 4 years).

It's been my experience (as a person who directed game song groups within predominately African American communities in the Pittsburgh, Pennsylvania area and who in the late 1990s and early 2000s faciliated workshops on integrating music into the pre-school/day care environment) that few pre-school teachers or day care providers know many singing games. The only games two games that seem to be known among Black American and also among White American pre-school teachers, day care providers, and parents are "Ring Around The Rosie" and "Hokey Pokey".