My daughter is allergic to mosqitoes. She swells up horribly when she's bitten, they get red and inflamed, and depending on where she's been bitten, sometimes she can barely bend an arm, knee, or finger. We have been experimenting with lots of different natural remedies, as I refuse to spray her full of chemical bug sprays.
The one we've been using lately that seems to work well for her is straight organic vanilla. Yep, the same vanilla you use in baking cookies or flavoring cakes. But real vanilla--not artificial vanillin. Put it on the pulse points and/or all over the arms and legs, any exposed skin. Mosquitos don't like the smell. You might get a few brave souls who will still bite, but you won't get hordes of them feasting. It can be a bit sticky, but it isn't that bad. Of course, you can't go swimming, as the vanilla will wash off in the water.
I used it when I was outside working in my flower beds, hot and perspiring, going back and forth from sun to shade. I got three bites (but I'm not allergic, so they don't bother me). My daughter used this method last Sunday when she had lacrosse practice from 6-7:30 pm--when mosquitoes are just starting to come out and are active. And their field is low/lies in a bog area with not a lot of wind. She had zero bites when she came home.
If your little one does get bit, the sooner you can take the poison out/the venom they inject under the skin when they bite, the sooner the issues and problems go away. We have found that using meat tenderizer (like Lawry's) mixed in water and applied to the mosquito bite works wonders.
We have found that essential oil of Lavender works wonders on helping to take away itching--and it also relieves inflammation if you put it all over the entire swollen area.