What small organizations are actually able to do
A congregation or a community group cannot litigate a habeas petition, and does not need to. The things that reliably matter in these situations are almost all things a small organization is well placed to do, and they are frequently the things nobody else is doing.
Knowing where a person is. Families often do not, and the two official search systems in the United States are free and do not require a lawyer. Both are linked below.
Getting a message across a border. The Red Cross and Red Crescent movement runs a tracing service in almost every country for people separated by detention and migration. A family can start a request through their own national society.
Keeping a record. A note of the date, the office, the name and what was said, written down at the time, is worth more months later than anyone expects. Organizations are usually better at this than families in crisis, and it is a real contribution.
Money for the ordinary things. A telephone, a SIM card that can actually be registered, transport to a consulate in another city, a document reissued. These are small sums that decide whether anything else is possible.
Not overpromising. A person who has been told that help is coming, and finds that it is not, is worse off than one who was told the truth. Saying what an organization can and cannot do, plainly and early, is part of the help.
Who is already working on this
Several organizations have been documenting these transfers, litigating them, or supporting the people affected, and a group that is new to this does not need to start alone. The resource directory lists United Nations bodies, regional mechanisms, and nongovernmental organizations, with what each one actually does.