Makes me wonder if there's a different solution to the elevator problem entirely. Expand the scope to "moving people along a vertical axis efficiently". What would a safe continuous elevator look like? Or could the solution for going up be different than the solution going down?
There's the Paternoster[1], which is essentially what you're talking about, but as it probably fairly obvious, they're pretty difficult to make both safe and efficient.
I'm reminded of the 'Moving Platform'[2] idea proposed for high-speed trains fairly recently though, and I wonder if there's anything there that could be reused.
What would a packet-switched elevator shaft look like? (Lots of small cars, which divert to individual 'station' queues for each floor, and latch onto a continuous moving belt at fixed intervals - shades of ATM)