If received in a level and left untouched, a Dragon Star can be selected as a reward upon completing the level, and it will be transferred for free to the camp, where it can be merged.
It is possible to gain one extra dragon gem, if the dragon star is tapped only once in the level. It will still apear as free reward to gain. Otherwise, you can keep in untouch and only tap it when it arrive to camp. So every 5 dragon star merge you can get additional 5 gems by clickingeach of them once before merging(if you didn't before), either at the camp or at the level itself. Be sure to click only once because dragon stars differ at their "tap times" before turning into level 3 stars, some have 2 "health", some have 3 and rarely even 4. However, you can't differ one from another(only slightly size difference) then for safety click any dragon star only once before merging.
Dragon Stars will not appear in Child Mode presumably because purchases with gems is also disabled.