We have some pretty detailed threads on this here, so it would be worth seeing if you can find them.
The sum of it is that GFCF by itself isn't really an answer, BUT there is a simple truth behind it: when kids with food sensitivities eat foods they shouldn't, they feel bad and act poorly. Diet CAN keep a child from being his better self, IF he has a food issue.
There are some different theories, some of which are easier to test for than others, and all that remains an evolving area. The net result is that IF your child has a food sensitivity, gluten is the most likely one. Many families here have felt healthier eliminating JUST gluten.
Each sensitivity needs to be considered separately. Unless your child is diagnosed with Celiac disease, going the full route is unnecessary and can actually be counter productive, since CF requires an increase in soy products, and we have several families who have issues eating soy (mine being among them).
So. My list of common sensitivities, drawn totally anecdotally from families on this board:
Gluten
Cassien
Soy
Eggs
Food Dyes
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Mom to an amazing young adult AS son, plus an also amazing non-AS daughter. Most likely part of the "Broader Autism Phenotype" (some traits).