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There is no procedural difference between a departing domestic flight and a departing international flight in the US, so your question would be the same as one for a domestic-to-domestic transfer.
Generally, checked luggage is checked through from one flight to another when you have booked it on the same ticket with the same airline, so you shouldn't have to claim your baggage.
There is an above-ground (not tunnel) post-security connector between TBIT and Terminal 4 at LAX, so you shouldn't have to go through security again. It should only take a few minutes to walk between the terminals.