There are some subtle differences between an intergenerational event and a family event. The family event encourages interaction between members of a particular family. Most events in the church are intergenerational, if not by intent at least by participation.
Worship can be an intergenerational event.
All are welcome to participate, but there usually is no intentional way for adults to interact with children, and youth do not usually engage in dialogue with adults. Fellowship dinners can be intergenerational, but do not intentionally provide opportunities for all ages to learn together. There are several events that with just a little tweaking could provide all ages significant opportunities to share a learning experience together. It is appropriate at various times during the year to invite children, youth and adults to participate in each other's class on a given Sunday morning. Moms and Dads can spend a Sunday in the classroom of their children at the invitation of the teacher.