Playing with kites comes under the general basic shar‘i principle having to do with games, which is that if it is not a game that is forbidden, such as playing with dice, then it comes under general stipulations such as the following:
1. It should not distract one from remembering Allah and praying
2. It should not distract one from doing some essential duty
3. It should not involve extravagance, because that is blameworthy
4. It should not involve any haraam betting between competitors. For more information on that, please see the answer to question no. 120221.
5. In the case of kites in particular, they should not be in the shape of animate beings or animals or birds, unless the one who is playing with it is a small child, because there is a concession for small children that is not made in the case of adults.
See also the answer to question no. 22305.
And Allah knows best.