Hi, is there any way, how to isntall the DFI Fly App on a children's phone with Family Link? I've tried it on several phones and always the same problem.

Same here its not working on my phones.

Open Family Link App or parent's phone.  Select child's name that you are trying to load the App on their phone.  Tap Controls. Now you can Tap on the name of the child's phone.  Scroll down and allow Apps from unknown sources.  Accept the warning.  Go back to the child's phone and enable Apps from unknown sources in the settings.   You should be able to install the App now.  

I have the Family Link app on my Samsung Galaxy S24 Ultra and installed the DJI Fly App on a Samsung Galaxy A15 for my grandson.  

Hope this helps you also.

So frustrating. Unfortunately your advice didn't help. Is the same ad use plastered all over the Internet. However, on my family link, once you select controls and the phone or visa versa (can be done both ways. You get the attached screenshot with not permission option for install from unknown or 3rd party sources.  At this point my options are to return the drone and purchase from a non Chinese blacklisted company. (I work in government GIS, this is a big problem), or remove all the restrictions and let my 13 year old free on the Internet to watch youporn while cooking meth and 3d printing guns.