I noticed a lot of people did not size down their bands, by pulling off the outside strip. Even for small children! Some of the babies I saw with bands wrapped around their wrists looked like they were going to lose them at any moment.
I think having the long strap sticking out would make it a lot more likely to come off. I sized mine down, and had no problems at all. Nor did my husband, who has bigger wrists than me and didn't size his down.
I left mine on all the time, even sleeping. He took his off whenever he got back to our resort room. It didn't seem to have any effect on how securely they stayed on.
I strongly recommend making sure they fit properly. Also, if you're worried about a particularly active child, make sure to bring a small rubber band to add extra security (or buy one of the sliders). Even tape or a bandaid would do the trick.