Yet one more way to look at this, which is also consistent with the employer “paying for business in order for the employee to arrived rested and ready to work” is that if the employee has decided to go out-of-pocket to fly his family with him, it could well be that the best way to accomplish is his arriving having had some quality time with his children on the flight and a well-rested wife who presumably will need to look after the children whilst he’s working.

He should not be required to leave his family behind, and seems willing to pay to bring them. Again, unless he’s underperforming, leave it to him to decide the best way to accomplish that.

Just because an employee is on a work trip, doesn’t mean he’s the employer’s property 24/7. Figuring out the best way to get the job done and manage his family should be up to him. If it’s not, the employee might want to consider applying for work with a more understanding employer.