I live in a grammar school county. Children do indeed move to the grammar schools at 12 and 13 and for 6th form. Very few apply though. The majority of our secondary moderns are good schools. A few are not. One is outstanding. Loads and loads of the children going through these schools into their 6th forms then go to university.
The money follows the child. This is standard procedure and taken account of in the census of pupils each year. There are no budget problems caused by transfer to grammar school, however, because, in effect, hardly any children move. Far more move due to parental relocation.
The least favoured grammar schools (there is a pecking order) can usually squeeze a few more in. They also take from far and wide and do not stick to catchment area. If entry is at 12 or 13 plus, the grammar school sets the exam not the LA. No-one transfers at Y10 or Y11. Grammar schools are not allowed to keep places open for late transfer. They accept everyone who passes the exam for entry in Y7 and do so, as far as I am aware, but it may not be the first choice grammar school. However, not all cohorts taking the exam are the same and some years they do not fill up. They do not lower the pass mark, to keep standards up, but then can take children who "blossom" a bit later.
In the case of schools not having a 6th form, of which there are few here, they can also transfer to the secondary modern 6th forms who nearly always have places.
People who post on these threads have never been near a secondary modern. They have no idea that A levels are available or that children get 55% and upwards of GCSE's at A*-C including Maths and English. If any of you have time, take a look at Waddesdon, John Colet, The Misbourne, The Amersham School, Chalfonts and Great Marlow School and you will see schools you will not recognise as secondary moderns. Some have in excess of 30% high abilty children so why would they not offer A levels and academic GCSE's? It would be failing the children if they did not.
As for the bitter disappointment of failing the 11 plus, the Review process often offers to candidates who score one mark below the pass mark. Frequently children get in who are 3, 4 and even 5 marks off the pass mark on appeal . Some people do not appeal because they are happy with their allocated secondary modern.
The big problem is closing the gap in our secondary schools. This is being addressed by Pupil Premium funding. Hopefully, there will be improvement. I have met very, very few parents who have transferred out of the secondary moderns to the grammars. Usually people who are rather "stuck up" but not affluent and do not see themselves, or their children, at the same school as ordinary children. They cannot afford private, so they agitate to get their children into the grammar schools. I have seen children (two near me) pushed to breaking to achieve 12 and 13 transfers. (Bed wetting, bad behaviour etc). Most people do not do this to their children and are quite happy with the schools they are allocated.