Having completed a path for the balls to arrive at the minute rack the trick was to come up with a way of doing it different than others I have seen.
I started with the hour rack because it had the most balls and if I could work that one out, it would be time to start all over. As I said earlier, one of the challenges of this project was to fit everything into the dome. As it turned out, I had just enough room to rack 12 balls.
The hour rack is unique because at 12:59 and one minute, there has to be one ball left in the rack to represent 1:00. Conventional ball clocks will capture one ball in the rack so it cannot be released with the others. 
This proved to be a challenge because of the way I chose to rack and release the balls. This first photo shows the hour rack on the bench. To the right between and above the first and second ball is a blade which when the balls are released, the blade will keep one ball from exiting the rack.
This all happens when the hour ball for 1 o’clock tips the lever to release the balls.
The other racks a pretty straight forward. The ball that trips the rack will continue down to the next level.
With the 1 o’clock trip ball it will be necessary for it to return to the lift.
I mounted the first rack on the bottom and worked up from there just to make sure I had enough vertical room to return the 1 o’clock ball.
The trip of the rack occurs when there is a ball in the full position. That pushes back a latch that had been keeping the release from tipping when a ball passed through the tube on the right end. I.e. 9 for minutes, 5 for tens of minutes and 12 for the hour.
The balls will have to return to the lift once they have exited the rack.
Again the limited vertical resource came into play as I had to make sure there was just enough drop for the balls to exit the rack without over doing it.
One ball is not an issue but when you have 12 stacked against each other, there is friction against them moving to consider after they have been stationary for 12 hours.
Racks are mounted, but the balls have nowhere to go.
This last photo shows what 12:59 would look like.
This video shows the balls cycling through the racks. They still have nowhere to exit, so had to improvise. The 1 o’clock ball is captured and I have to manually release it to go to the left end of the rack. That will come next.