Last Train to Nowhere is a challenge that takes place in Lynchwood. On the main street there is a digital clock tower. Between 05:00 and 06:00 a train will pass through the main street with a bandit standing on its back platform. The train passes periodically, about every hour-and-a-half in-game, but only the 5 o'clock train will have the bandit on it. To complete the challenge, a player must kill the bandit before the train disappears.
It seems that the bandit disappears with the train even when knocked from it. The siren's Phaselocking will lock him in place, though, making it much easier to kill him before the train disappears.
If the bandit happens to spawn as a brute, his durability coupled with the train's speed will make this a very difficult challenge to accomplish. This can be mitigated in Normal Mode cases where high level characters above 30 have a significant damage advantage over enemies capped at level 30.
Do not try to ride the train. In a game where you can fall from tremendous heights and suffer no damage, you might reasonably assume that you can jump onto and ride a moving train. You cannot. The train will instantly kill you.