The eczema response would be an appropriate response, although I always thought eczema was more of a dry skin/scaly/flaky type of skin disorder. A creamy moisturizer, in that case, would work. Psoraisis (sp) would be another thought, but it sounds like she doesn't have patches of extreme dryness or cracking.
Six years old is awful young, but children are also known to get what's called (acne) rosacea. In this case, though, the bumps are not acne/blackheads/pimples/
whiteheads, etc. and are rounded, rather than coming to a "point." If it is rosacea, the normal "acne" treatments can irritate and make the rosacea even worse. I know, because I have rosacea and am trying to recover from a severe flare up brought about by some detoxing I've had to do.
I would do some Internet exploring on these three types of dermatological problems and see if any of the symptoms/photos look like what your daughter has. Another thing that can cause a skin outbreak would be if she's sensitive to any foods; the body's defense in trying to detox or rid the body of the food it doesn't like could very well result in a skin outbreak.
On another subject...
My daughter had what I thought might have been chicken pox, as she was starting to get red spots all over her body. (She's had the original and the booster varicella vaccine). However, the spots weren't blistering in the middle, like the chx pox does. So, with the clinic closed at noon yesterday, I had to take her to Urgent Care to find out a specific diagnosis.
She has something called Pityriasis Rosea, which I have never heard of before, nor have I ever known anyone who's had it. Apparently, though, according to the doctor, it's fairly common in the 6-10 year old bracket (and all the way up to 30 yrs old). Instead of getting the characteristic blister in the middle, like a chx pox does, these spots get a dry skin center, which can flake off.
This Pityriasis Rosea also acts like chx pox in that some children get a very mild form, while others get a really extreme case. Researchers still do not know what causes it (although there's speculation it could be a mycoplasma, picornavirus, or the human herpesviruses 6 or 7), and there is no cure other than to let it run its course, which is anywhere from 6 weeks to a few months. It is NOT contagious, although it can get itchy. Best defense is to put a creamy moisturizer on each spot at bed time, to help alleviate the itching from the spots drying up, and give Claritin, Zyrtec, or Benadryl for extreme itchiness.
Sigh. My daughter was one of the few babies the doctors had seen where the little red spots/dots from Roseola were so very pronounced; we thought she had chx pox then, too.
Fifth disease, hand-and-foot, Roseola, Pityriasis Rosea -- I don't remember ever hearing about these disorders, or even knowing someone who'd had them, when I grew up (I'm dating myself when I say it's been 40 years...). Makes you wonder how come all of these disorders are now becoming so prevalent.