Do your little kids wear glasses? Generally speaking, as myopes with moderate prescription have to wear glasses over a long period of time, they are liable to get sunken eyes. Furthermore, with the fatigue of the eyes, they are likely to manifest the symptom of sunken eyes due to metamorphic eyeball. Therefore, you should pay attention to your kid's ocular hygiene and ask them not to stay up and to relieve the fatigue of their eyes by dripping some eyedrops properly. And tell them not to wear glasses as far as possible, if they have to wear glasses over a long time, contact lenses are better for them. Besides, more diets which do good for the eyes should be available.
As far as I know, there are various causes that could give rise to sunken eyes with children. The most common reasons are lack of sleep, much too slim and pale, and other eye-related diseases. So, it is indeed a serious problem which deserved your due attention. My advice for you is to take the kid to consult an eye doctor in a good hospital and follow the doctor's directions. Speed is of the essence.