family members left at home
Refers to the family members who stay behind when someone goes away, e.g., on a business trip or assignment.
With the solo job transfer, the family left at home is just my wife and child.
留守番 is the act of looking after the house or the person doing it, while 留守家族 specifically refers to the family members staying behind.
Compound of 留守 (absence from home) and 家族 (family).