I have probably seen too many movies, but how feasible would it be to have the last car on the train be the one that stops in each city?
When you board your train you will select the car that stops at your destination and it will decouple and coast to the station like a roller coaster.
To account for people wanting to get on half way, I can envision two different scenarios. The first method would involve the original train never slowing down, only losing cars. New trains getting up to speed would create enough of a buffer to alleviate safety concerns.
My other thought would be to have the trains recouple at speed like refueling aircraft at altitude. If the cars departing the station leave before the train reaches the station it could be up to speed and The train could separate so the new cars could join in the right spots to fall off of the end at the appropriate stops. Otherwise the cars could be added at the end and the passengers could relocate between stops.
Regenerative Braking could be used to add power back to the system at the stations if other trains are leaving, reducing load on the rest of the system while one train is accelerating.