When they accept the invite, they're added to Premium for Family straight away. They won't be charged for your account any further - Their payments are handled by the plan owner (you). Any Premium time they have remaining will resume if they leave Premium for Family.
Note: If your kids purchased the Premium subscription via a gift card or e-card, they'll lose any remaining Premium time when accepting an invite to Premium for Family.
However, if their Premium subscription is currently handled by a third party - such as iTunes or your mobile operator - they'll need to contact them to cancel their Premium first.