In the United Kingdom, a Supreme Court case has established that the court, in extradition cases, must take into account the impacts of extradition on children. Obviously, there's always an impact, but it can't simply be that the child is deprived of the care of their parents. It has to rise to a more severe level before the court will say it's sufficient to bar extradition. For example, it might be there are no care arrangements in place for the children, because both parents are being extradited. That would leave them with no one to care for them. Even then, the Crown will look at whether there are alternative care arrangements that could be made, through foster parents or other types of things.
The other types of issues related to children that can lead courts to find it's not proportionate to extradite are medical conditions and mental health issues—something that rises above the ordinary. I'm using that advisedly, because, obviously, it's terrible for children to be separated from their parents. However, the court has to be satisfied there is something more than what would normally be the case, were a child to be separated from their parent because they've been extradited.