I constructed a thought experiment, can't figure the answer.

Two long (say $1\ \rm{km}$ long) railway cars are on parallel east-west tracks in a zero gravity environment. Each car has a light source at one end (say the west end) and a device to measure the speed of light at the other end (the east end). The cars are started a long way apart, accelerated toward each other then allowed to coast at the same speed but opposite directions until they are next to each other. Mechanisms on each car touch the other and trigger the light sources. [edit - the idea here is that by mechanisms touching the light sources are triggered at the same location at the same time in the stationary observer's frame of reference] The beams travel to the detectors and Einstein says each will register the speed of light as the constant c. [edit - and here is my error. As pointed out in one answer the one way trip can't be measured, so the detectors on the cars are useless]

I'm OK with that [edit - I was OK with that, not any more]. Here is the part I don't understand.

There is a stationary observer positioned further east. Does he see both beams arrive at the same time? Since both cars are travelling at the same speed relative to the observer but in opposite directions I would expect one to be delayed a bit. If they do arrive at the same time how does one reconcile this with the detectors on the cars saying the speed is constant but they would register when the two cars have moved slightly apart so the beams are not at the same place as they pass the detectors on the cars, so one must 'speed up' to catch up with the other when they reach the stationary observer.