At LAX, on the international arrivals terminal, (non US airlines), once through customs you can either turn left, up the ramp and out of the airport, or turn right to other terminals and there is a baggage conveyor to drop your baggage, which, in my experience, will have originally been labeled with a tag that will say both LAX and your final destination in the USA, I would imagine the other international terminals are similar.
The USA has a policy that all pax must clear immigration and customs at first port of call in the USA, some final destinations may well be domestic only and without regular on site immigration and customs facilities. All baggage will come to the carousel at the first port of call. I would tell original check in that I intended to break my journey at LAX.