Q: How do they end up in the kingdom after being brought to life?
A: Within the kingdom there is an organization known as The Guides. Members of this guild have devices which alert them to the magical surge of a Carousel Beast coming to life. Once this alert is out one guild member is alerted to be the individual to retrieve, inform, and guide the newly formed Carousel Beast. Their device then allows them to alter their own natural magic ability to teleport to their own old body to instead teleport to the old body of the newly made Beast. Once there their job is to explain what is going on to the new beast, explain how to use their innate ability to pop into Caroulot, and offer to guide them for a short time in this new life they have.
Q: Do they remember being a carousel or are they like a child when they begin their life?
A: They remember being a carousel and the children who most strongly powered their birthing.
Q: Do they have a specific duty to perform when they're created? What do they do with the lives they now have?
A: They don't have any particular duty though many choose to dedicate their lives to something related to the children. Some were loved strongly by those who were bullied and may go on to become guards or knights to defend the weak, others may have had tack that reminded children of nature so nature loving kids charged him and so he dedicated himself to be a naturalist. They are creatures with free will and have full choice over what they want to do with their new found lives but are often swayed by their children to what they want to do. They do pretty much whatever they want with their lives. Carousels have been magical things to children sense medieval days and so such an old special thing was bound to start collecting magical energy. There is no real rhyme or reason for them to come alive other than a strange buildup of energy