Oh, just seen this. I can answer the 'BUT HOW DO THEY KNOW?'.
Border Control are understandably very well trained in detecting nervous behaviour. You are 'logged' into and out of the country so your normal travel patterns are known.
They do not 'know' who has permission and who hasn't. They assess each person's demeanor and the more experienced they are: the better they are for judging you.
Therefore if you've never been stopped for taking your kids on holiday without full permission from the other parent, if they have shared-PR, this is largely because you didn't realise you needed it: your body language being less than nervy, wouldn't signal anything is amiss: after all, you don't know that you're doing anything wrong.
And secondly it's because nothing has flagged up e.g. the left behind 'parent' hasn't asked the courts/police for an 'all ports' notification.
I think in Karen's case, she would have been very unwise to take the children to their country of origin, as Toad theoretically could instigate a charge of abduction. Whatever his motivation for doing so, he could. The children are domiciled (e.g. live and schooled here) here until the courts give permission for them to leave, irrespective of what nationality they hold.
However, if you are knowingly defying the 'wishes' (nefarious, ambivalent or otherwise) of the other parent, you are running the risk of being accused of child abduction if you are caught, especially if you are visiting their/your country of origin.
The chances of being caught are low - but the consequences can be very severe, including loss of parental responsibility.
You decide whether it's worth it.