Yes. Once admitted in L-2 status, a spouse is authorized to work incident to that status. Derivative children may study, but they need an independent lawful basis for employment and cease to qualify as derivatives at age twenty-one.
The spouse's position is the strongest part of this route
Hypothetical example: the director's spouse has driven for the Coalhurst carrier for a decade and expects to keep working. Once admitted in L-2 status, employment authorization comes with the status itself, which is a substantially better position than dependants hold in some other categories. What that authorization does not do is provide a commercial driver's licence in the destination state, medical certification or endorsements, all of which are separate regulatory matters with their own processes.
Look at the children's ages against the assignment length, since a seven-year maximum outlasts a fifteen-year-old's derivative eligibility. Check every family member's citizenship individually rather than assuming it matches the principal's, because that fact can change the procedural steps a person faces.