An L-2 spouse is employment-authorized incident to status, which is a materially different position from a TD spouse. He should still hold documentation confirming his status for employment-verification purposes.
L-2 is the exception among dependent categories
Hypothetical example: Spouses in L-2 status are authorized to work by virtue of the status itself rather than by a separate grant of permission, and current policy treats the admission record as evidence of that authorization. Practically, an employer completing employment-verification paperwork will want documentation that clearly shows the category, so the admission record should be checked for accuracy on arrival. Children in L-2 status may attend school but are not employment-authorized. This is one of the points where the choice of classification has real consequences for a household, because the same family with the principal on TN instead would have a spouse who cannot work at all. That difference is worth weighing before a route is chosen for other reasons.
Family planning should identify each person's intended work, study and travel dates before dependants apply. Hypothetical example: A coffee-roasting company transfers its operations director, whose spouse intends to accept a teaching contract. The spouse's L-2 status can provide work authority incident to status when the person holds the qualifying status, but the family should retain current evidence of that status for the new employer. A child's age can make timing independently important.
Keep passports and relationship records valid through the planned travel period. A dependant's independent application issue can delay a carefully scheduled transfer.