"I have been feeling very upset about the fate of people who will have to remove their children from schools, and move away from their support networks, families and everything they know and settle somewhere completely different with potentially not enough money to feed their children."
The problem is, the alternatives are all hideous.
If you index benefit with living costs and number of children, which doesn't of course happen with wages, then you create disincentives to work which become greater with the number of children you have. Someone with four or five children who is unemployed is highly unlikely to get work which provides the same amount of money. A situation in which people who are already struggling to cope are essentially encouraged to have more children isn't good for anyone, least of all the children.
On the other hand, if you cap benefits at a level that will support an average sized family, people who have for whatever reason already got larger families sink into utter destitution, and you set in place some hideous mechanisms which make the 1834 Poor Law Amendment Act look like a summer school project by the Child Poverty Action Group.
I'm with Stalin on the Catholic Church (how many divisions have they got?), but I can't imagine that a proposal to make third and subsequent children born after 01/01/12 (to give people time to sort out contraception) not being eligible for benefit would exactly go through smoothly.
So what do you do? Salaries don't pay attention to family size. The tax system could, but doesn't. Benefits do. Working families have incentives to use contraception, non-working families don't, and in some cases have active incentives to have more children. Outcomes for children in poverty are bad and that's exacerbated by large family sizes.
Before the second war, both temperance and contraception were seen as socialist issues, protecting (largely) women from the consequences of drunken husbands drinking all their ages on a Friday night and of unconstrained childbearing. Marie Stopes was with hindsight a hideous eugenicist (she cut her son off without a penny for marrying a woman a wore glasses, even though she was Barnes Wallis' daughter) but the basic contention that giving women control of the sizes of their families benefited both women and their children was absolutely right. Unfortunately, post 1945, most of those arguments are so horrifically contaminated that no-one dares to talk about them. But if you could manage to, in an enlightened and progressive way, "nudge" people to have only the children they can afford to raise, with the state providing funding for up to two for reasons of common humanity, that would be a good thing. Encouraging people who are struggling to provide for their existing families to have yet further children is bad social policy, except for all the worse ones.