Dawn uses her "catchphrase," so to speak, only when she herself is nervous or to calm somebody down. This is mainly in situations where Dawn is competing against other coordinators or trainers that appear to be intimidating and powerful.
Iris, however, seems to say her catchphrase in almost every single one of her appearances. With little else for the writers to go by, Iris has been relegated to Ash's fault-picking coach. Iris always attacks Ash for being naive, overexcited, anxious, or making a crucial mistake. Iris had also called Trip a "kid" as well. This goes to show that Iris would simply call anybody a "kid" if that person was to do something she didn't approve of.
I actually loved it when Dawn said "No need to worry," because it is something an ordinary person might say to calm themselves down. Dawn was expected to fill her mother's shoes as a Master Coordinator. It had been established in early Advanced Generation that if a coordinator is nervous, the pokemon can sense this and not perform to its best ability. Dawn clearly went through a lot of stress as she evolved from a beginner coordinator to almost a Top Coordinator.