Mrs. Bute quickly gets the upper hand at Miss Crawley's house. She has spent her life ingratiating herself with Briggs and Firkin, so they are happy to have her there and serve her. Rawdon, meanwhile, has spent his life treating them like the servants that they are, so no one there is going to go to bat for him. The narrator pauses here to give us a cynical little moral lesson: try to be nice to anyone who might be of use to you later in life.
First things first. Mrs. Bute makes a big deal out of how sick Miss Crawley must be and imposes a restrictive health regimen on her. This mostly consists of Mr. Clump's medicine and lots of indoor time.
Also, Mrs. Bute forbids Becky or Rawdon from entering the house.
Regretting that her own family is too dumb and boring to entertain Miss Crawley, Mrs. Bute instead starts to badmouth Rawdon and Becky as much as possible to get Miss Crawley to stop liking them.
It's pretty easy to do. Rawdon has killed a couple of guys in duels, has seduced and abandoned a bunch of girls, and is a gambler. Becky is not only the daughter of an opera singer, but she herself has been an artist's model and even used to drink gin with her father. They're kind of made for each other, right?
Everything would be OK for Mrs. Bute except she's a little too tyrannical. Miss Crawley secretly hates her and doesn't want to be kept as an invalid prisoner.
Mr. Clump and Dr. Squills have a very unemotional conversation discussing the likelihood that Miss Crawley will die really soon under Mrs. Bute's oppressive "care."
Mr. Clump tells Mrs. Bute that Miss Crawley's life is in danger, and Mrs. Bute freaks out because Miss Crawley has yet to change her will: if she dies now, Rawdon will get all her money. So Mrs. Bute relaxes her iron grip.
Miss Crawley starts to go out in her carriage.
In the park, her carriage rides past Rawdon and Becky's, but Miss Crawley totally snubs them. Mrs. Bute one, Becky zero.
Mrs. Bute decides to take Miss Crawley to Brighton, a seaside resort town.