Yes. L-2 status and its employment authorization incident to status depend on her holding L-1 status of either kind, so the change from L-1B to L-1A continues his authorization and extends the family’s horizon to the seven-year limit.
L-2 follows L-1, whichever letter
The spouse and unmarried children under 21 of an L-1 transferee hold L-2 status regardless of whether the principal is classified as L-1A or L-1B, and an L-2 spouse is employment-authorized incident to status. In the hypothetical, her husband works as a farm-equipment mechanic for a U.S. dealership on that basis; the change of her classification does not interrupt his authorization, though his employer may ask for an updated admission record after her new approval and next entry.
The family’s time horizon changes with hers: under L-1B their status would have ended at the five-year cap, while under L-1A it can run to seven years, subject to each extension being granted. Their daughter, aged 12, may attend school and has nine years before the twenty-first birthday limit, comfortably beyond the cap. Changing to L-1A does not by itself establish eligibility for the multinational-manager immigrant category: that category independently requires qualifying managerial or executive employment abroad and a related U.S.
petitioner that has been doing business for at least one year. Should her status end early, the husband’s work authorization ends the same day.