When you sign your kids up (upon boarding) you list who is allowed to pick them up and you give a password to be used when picking them up (and they do ask you everytime you pick up for that password).
If they are 7 & under, you or a person on your designated list must check them in and pick them up. 8 & above, you can tell the club that you authorize them to let you kid come & go in their own, but you decide this. The dinner time can vary a bit, but it's usually around 5:30 or so. When you register them you get an entire written schedule for the whole duration of your cruise (a kids room Navigator), so you know every activity and meal and the times for each for the entire cruise. That way you can plan ahead when to drop the kids off or pick up