A parent should start with the return, not with an application for every family member. The W-7 instructions require a dependent claimed for an allowable benefit to appear on the attached return, with the relevant schedule or form. Some family benefits require an SSN and cannot be obtained simply by requesting an ITIN.IRS
Having an ITIN does not itself prove that the person qualifies as your dependent. The relationship, residence and other tests belong to the tax return. A child who must file their own return has a different application reason from a child included only for a parent's allowable benefit.IRSPub. 519
For a non-resident parent filing Form 1040-NR, dependent claims are more limited. IRS guidance covers residents of Canada and Mexico, U.S. nationals, residents of South Korea under additional conditions, and qualifying Indian students or apprentices. It does not give every non-resident filer the same dependent rules.IRS