That is funny - you say "less and less casual" - while most people would say the opposite, I believe.
This is from the RC web site:
There are three distinct types of evenings onboard: casual, smart casual and formal.
Suggested guidelines for these nights are:
Casual: Sport shirts and slacks for men, sundresses or pants for women
Smart Casual: Jackets and ties for men, dresses or pantsuits for women
Formal: Suits and ties or tuxedos for men, cocktail dresses for women
The number of smart casual and formal nights is at the ship's discretion.
Generally, most nights are smart casual, formal occurs every three or four days (twice on an seven night cruise).
The rest are casual.
Technically kids are given leeway. If you feel personally compelled to follow the rules then here they are - however if a child does not adhere to the rules he will still be allowed in as long as he is with properly dressed grownups and not just plain sloppy looking.
For formal night - I have not seen a ship where a tie is mandatory for years. However they are recommended. A collared shirt, preferably not a polo, with a jacket is highly recommended.
Now, I don't have kids so I hope someone who does will come along and answer this.