You can certainly include your adopted chidren when covering for dependents under your travel insurance policy. Typically, its not clear how they would know the difference unless they do have a different last name which might be the case in a second marriage. Rest assured however, that your adopted children can be covered under your policy.
The adopted child can be instantly covered for up to 31 days from the time they are placed in your home if you already have a travel policy. You would then need to call the insurance after the 31 days is up and renew the coverage to ensure that the child is covered. They may ask for the paperworks so keep all of your files handy. This is the best method when shopping for the cheapest annual travel insurance.
You may also add eligible dependents that the court has ordered into your policy. This may incur if there is a child in question, and possibly in the case of an adoption in process, so you might want to ask what is required. In instances where there is a court order that a child should be included, then they are included but only for up to 31 days, and when that 31 days has expired they have to be renewed.
If you wanted to include an extended family living with you, some insurance companies would allow this so long as they are really under your care or under the care of the person applying for the insurance. This could be a grandchild or perhaps a niece or nephew. The insurance includes annual family travel insurance. The documentation that the insurance company is really interested in is the documentation that states that the covered individual has legal guardianship of the child. Understand it carefully because there might be loopholes that would state that a child could not be covered if they are on someone else’s insurance policy.