The same logic applies. A spouse may qualify for TD regardless of nationality, with the procedural route set by their own passport, and will generally need the marriage certificate as relationship evidence. TD does not authorize the spouse to work in the US.
Plan the spouse's steps on their own track
If the spouse needs a consular appointment while the children do not, the family may not be able to travel as one group, and planning around a single move date can fail on that alone. Decide early whether the principal travels first. Also settle expectations about employment: a spouse who needs to work will need a separate basis for it, which is a question for counsel before the move.
Two parties control this timetable and the family is neither of them: the consular post that schedules the appointment, and the employer that sets the reporting date. Between them sits one decision that belongs to the household alone, which is whether the principal goes ahead without the others. Agree who will tell the employer if the spouse's appointment falls after the start date, because an employer informed late generally has fewer options than one informed early.
On employment the constraint is a matter of status rather than willingness: TD does not authorize the spouse to work, so a spouse who intends to earn needs a separate basis, and identifying that basis is work for counsel before the move rather than after it.