This might not help with the heat & hot sweaty feet, but with hot spots from wearing tennis shoes all day & walking, walking, walking:
Wear cotton socks & carry with you a spare pr of socks for each of you. Blisters might be avoided if you can change damp socks during the day .. carry moleskin or bandaids just in case. Maybe little feet don't sweat as much as adult feet??? Make sure the tennis shoes fit correctly & are 'broken in' before the trip. Also make sure they have a good arch support!
If you have a second pair of tennis shoes, alternate days with them. (rubbing from shoes=sore feet)
My actual solution costs $ - buy a pair of Tevas for each of you...they are so comfortable, (good arch support) & very durable & can be worn with socks (on airplane ) & without (at parks), can get wet (pools/waterparks)
Guess it depends on how many days you will actually be doing so much walking - if it is worth the purchase. And also if you live in a climate where you can get alot of use out of them before they are outgrown (but you can pass them on to the siblings)
They are probably on sale at a store near you...But I wouldn't wear any ordinary sandals just to keep cool, I think achy feet/ankles are worse than warm ones!