To overcome the lack of cars in an elevator shaft you build two shafts, one for the cars to go up in, and one they come down in. They can be attached in a chain like arrangement, so it's an "all stops" affair. All the cars move one floor at a time. When a car arrives at the top it moves across to the "down shaft" and now moves down to the basement again (No, the car doesn't need to turn upside down ;-) only the movement mechanism needs do that, the car can "rotate" so it's always the one way up.
This was the original design for an elevator system. It's problems include the time taken is proportional to the number of floors traveled, and you can't easily block off floors. But it does provide for a much larger volume of traffic.