Of course, social mobility needs movement down as well as up and that is one of the reasons it's stalled now.
Fictional example:
Mary comes from a family where everyone went to elementary schools until they reached school leaving age and then they started work. Mary is the first one with the good luck to be born in the era of free secondary education. She passes the 11+ in 1960 and goes to grammar school. Her family are not all that well off (father a garage mechanic, mother a shop assistant) but they can afford the uniforms and the fares and they want their children to do well, so they seize this wonderful opportunity for Mary.
Mary works very hard and does extremely well. She stays on well past the school leaving age (15 then) and takes O levels and A levels. She now reaps the benefit of the other huge change in education and goes to medical school - no tuition fees to pay, generous maintenance grants. In due course she becomes a consultant obstetrician and marries John, who has had similar good fortune in the education system (father a miner, mother a cleaning lady - John has qualified as a solicitor).
John and Mary have a daughter, Claire, born in 1989. They choose to live in Inner London because it's convenient for work. However, it's not so good schoolwise. They send Claire to private schools all the way through. Claire is not as high-achieving as either of her parents, nor as driven. They pay for her to have private tuition so she can keep up. She gets reasonably good GCSEs and A levels and finally gets into a Russell Group university where she studies English and has a lovely time. When she graduates with a 2.1 (just), her parents fund her through a gap year travelling, and then a series of unpaid internships and low-paid jobs until she is finally able to get a job in publishing when she is 24.
Now, this is fictional, but there are lots of people like John and Mary. They have (entirely understandably) done their level best to keep Claire from falling down the social ladder. But the inevitable result of that is that their own counterparts in Claire's generation don't get the opportunities that they had a generation ago.
The Sutton Trust is doing its best to draw attention to this kind of thing. The middle classes have a stranglehold on the best state schools. It is a lot harder now for children from poor families to move up the ladder than it was 40 years ago. This is depressing.