Both can hold L-2 status as unmarried children under 21, and both can study. Neither can work in L-2 status; only the L-2 spouse has employment authorization. The 19-year-old will lose L-2 eligibility at 21.
Two years of eligibility for the older child
L-2 status is available to the spouse and to unmarried children under 21 of an L-1 worker. The 17-year-old can attend high school and the 19-year-old can enrol in college in L-2 status without a separate student visa, although some institutions ask L-2 students to change to F-1 for other reasons. Work is a different matter: employment authorization incident to status applies to the L-2 spouse only, and the children may not take jobs, including campus jobs.
The 19-year-old's L-2 eligibility ends on his 21st birthday, which on a three-year assignment falls inside the stay. Before then he will need his own status, most often F-1 as a student, or he will need to leave. Plan that change well ahead of the birthday, because a change of status must be filed before the current status ends.
Bring both children's birth certificates and confirm the marriage record for the spouse. Hypothetical example: A packaging-automation specialist will transfer with a 20-year-old child who is finishing a professional certificate. The first review should calculate the child's age at travel, the expected end of the L-1 assignment and the date derivative eligibility ends, then identify a school adviser who can explain an independent student option.
The birth certificate and current admission record settle the family relationship and status, while the institution's rules settle enrollment. Do not assume the child can work to cover tuition while dependent status continues. A late change can interrupt study even though the principal worker remains employed.