Those are all great examples.
One more example that I can think of, because I have recently gone through the whole process, is to use: "to put somebody down". This is used for children who are quite young - no more than infants. You probably wouldn't use this expression for toddlers older than two or two and a half or so. Not only would you use it for that age of child, you might restrict its use to when the child is already asleep in your lap, or is very close to being asleep. "Putting them down" describes the entire process of getting up and taking the child to bed and putting them down into the bed for them to continue sleeping.
"Honey, give her to me. I'll put her down. Where's her pacifier?"
"Jake, be quiet! I just put Michael down. You'll wake him up!"
Be aware that, depending on the context, it can have the same meaning as "putting a pet to sleep" because of illness or old age - like they did to Marley.
"Mom, where's Buster?"
"Oh, honey, he got hit by a car and they had to put him down. He's in a better place right now."
Hope this helps!
Best to all,