My ankles are quivering at the very idea of hiking with a kid on my back in just sandals. I know, I know, they’re cool and can feel comfortable. But they offer absolutely no ankle support, little foot protection, and next to no edging control on steep, slippery sections of trail. Trail sandals is a growing footwear category, but I just think wearing sandals while hiking with any kind of a load is a bad, bad idea. In particular, if I were carrying a child on my back, I would want to wear the sturdiest, best-traction boots available to ensure that I don’t stumble when a sharp branch digs into my foot or something like that.
Still, I empathize with your desire to have shoes that breathe well on warm days. Scarpa’s Vento Mid ($130) is a good, light boot, with over-the-ankle coverage and fabric-and-leather construction that offers reasonable support and good breathability. A similar boot with slightly taller construction is the Chameleon Ventilator from Merrell ($110). It’s designed with hot weather in mind, and has special breathable mesh panels on each side, along with suede leather for durability. Lastly, Boreal’s Midi ($140) is a beautifully made light leather hiking boot that has a wicking liner to help keep your feet cool and dry.
Any of these would work well, I think.