Kids can't sign legal documents. How do they hold them for 7 years.
There are laws for their parents and guardians to follow and no child actor signed to a contract is without a lawyer. That isn't to say greedy parents aren't signed on as 'managers' who get their cut but that the whole shebang is regulated.
Google: Jackie Coogan Law
In general they stay voluntarily. Most adults aren't signed 7 years in advance either, I think. It's usually 4 years these days. With kids you get even less security, but people like the job.
I remember the Harry Potter kids were signed on for every movie (movie yes, but still) individually. Emma Watson had to be talked into quite a few of them in the middle of the series.