I would have though it meant something along the lines of “Mammaí sent the child up (stairs)”

My guess is that it’s more along the lines of “Mammaí put the child to bed.” In American English at least, when you actually take your child, especially an infant, and physically assist in putting them into bed, you could say that you put them to bed, and if the bedroom is upstairs, you could say you “put them up (to bed – for the night).”

The other examples with the cat on pages 68 and 69 would also imply physically assisting the cat in going outside (putting the cat out).