The spouse is employment-authorised incident to L-2 status and may work. Dependent children may study but not work, and dependent status ends at twenty-one.
One adult's status determines what the other may do
An L-2 spouse holds employment authorisation by virtue of the status itself, so the spouse may take a job without waiting for a separate grant of permission, although an employer may still ask for evidence of it. That freedom is one of the practical reasons households prefer L over TN where both are available. Children admitted as L-2 dependants may enrol in school and are listed among the classes permitted to study, but they hold no work authorisation, which matters where a teenager expects summer employment.
Dependent status ends at twenty-one, so a child arriving at eighteen for a five-year assignment needs an independent basis, typically student status, before that birthday. Because L-1B itself is capped at five years, a household should read the assignment length, the children's ages and the cap on one page rather than separately.