TD status does not authorize employment of any kind, including remote work for an employer located outside the United States. An eligible spouse or unmarried child under 21 may seek TD, but a spouse's own work plan needs a separate authorization or category assessment.
Treat every household work plan as its own question
List each planned activity, including remote work, freelance work, or a future job search, rather than assuming a TN professional's status covers the household. Children's schooling and any upcoming age-related deadlines should also be recorded. If arrivals will be staggered, note which documents each person needs at the time they actually travel.
Dependants in a dependent status are tied to the principal's status remaining valid, so a gap in the principal's authorisation can affect them as well, and any renewal should be timed alongside the change rather than after it. School enrolment often carries the earliest deadline in the household and the least flexibility, so identify it now and treat it as the fixed point around which other dates move.