No. A child admitted in L-2 status may attend school but is not employment-authorized on the basis of that status. Only an L-2 spouse is authorized to work incident to status.
The spouse's authorization does not extend to children
Hypothetical example: the specialist's family includes a spouse and a nineteen-year-old who plans to study and hoped to take a part-time job. The spouse, once admitted in L-2 status, is employment-authorized incident to that status. The child, by contrast, may study but may not work on the basis of L-2 status alone; any employment would need its own lawful basis, for example a student route with its own work rules. The child's age also matters for the length of the plan: derivative eligibility ends at 21, so a five-year L-1B assignment would outlast the child's ability to remain as a dependant. Each family member's own activity determines the route. Confirm citizenship for each person, because a family member who is a Canadian permanent resident rather than a citizen may need a visa stamp even where the principal does not.
Hypothetical example: a jewellery appraiser accompanies an L-1B transferee and plans to accept appraisal assignments from former clients. A qualifying L-2 spouse may work with authorization that accompanies valid L-2 status, but an employer or client may require proof of status and any occupational credential. Do not assume a foreign appraisal credential transfers to a regulated U.S. activity; check the state rule before advertising services. An unmarried child under 21 may study in L-2 status but cannot be employed. List each family member's intended activity separately, because the admission record, licence question, and age rule may point to different next steps. The principal's specialized-knowledge evidence does not establish a dependent's work qualifications.