Why do you want to check in at LAX for your LGW-DTW flight?

Checked bags depend on the airline. I've done JFK-DL-LAX-DL-SFO-SQ-SIN once. JFK-LAX was an evening flight. LAX-SFO-SIN was the following morning. The DL staff at JFK had to check my bag all the way to SIN told me that the computer would not allow them to check my bags with LAX as the final destination even thought I overnight in LAX. At the same time, they could only print the JFK-LAX boarding pass and not the LAX-SFO one.

However, they did tell me to go to the baggage claim at LAX to get my bags retrieved. Did it at LAX with no problems. In fact, my bags came out before the rest of the bags on the JFK-LAX come down the chute.

I've also had the following:
NRT-SQ-LAX-DL-JFK on ticket 1.
LAX-UA-DEN-UA-STL on ticket 2.

A last minute travel change required that I get to STL as early as possible. I planned to do NRT-LAX-DEN-STL and throw away my LAX-JFK coupon from ticket 1. At NRT, I told the agent what I wanted to do and got the bags tagged NRT-LAX-DEN-STL without a problem.