Well, I got the back crunchies again, so I'm not sure if I should spend more time than necessary sitting up at the computer. So, lounged on the couch reading this
long paper about the effects of moving on kids.
We already moved during what is called "middle childhood", and while there are problems with that move, the guy is doing better socially than he was.
I don't know how the models they used work, but they use the data they collected to estimate the effect of many individual factors like moves, income, low birth weight, and others on cognitive and social ability, which is assessed via those questionnaires that teachers fill out. The problem with this is that the same teacher is not filling out the questionnaire when the kid in the study is in kindergarten and when they're in fifth grade. Maybe in aggregate all of these consistencies average out?
I found it interesting that you can compare life changes with cash amounts using these abstractions:
Although the effect sizes of residential mobility were very small, they were generally a bit larger than a $10,000 increment in family income, which was associated with differences in children's functioning of 0.01-0.02 SD units across outcomes.
They also have an analysis that separates school moves from residential moves. The finding was that school moves affected cognitive ability, but not social skill, which is surprising to me because school is where most of the socializing seems to happen?
The tl;dr seems to be that moves have effects on development that can be ruled causal but seem to be very small effects, though they did not measure eighth grade social skills.