What is the proof of a family member's departure abroad (for registration of care)?

One of these pieces of evidence is the response to the request of the State Border Service.

Submitting a request for information on the crossing of the state border of Ukraine by an individual and on the presence or absence of a temporary restriction on the right to leave Ukraine or enter Ukraine for an individual is provided only upon a written request for information to the State Border Service (by mail to to the Ukrposhta system or in person to the public reception of the Administration of the State Border Service of Ukraine).