I'm researching building a CNC machine that has a working surface area of 2 feet by 4 feet. The purpose of the machine is to move to and push up an array of rectangular shafts in order to represent a 3 dimensional terrain. Others have made similar dynamic shape display systems, however my application need not be so dynamic, nor expensive. The goal is to be able to push up the pieces, one at a time to represent a rough interpretation of a terrain map (i.e not dynamically move based on motion sensors). What I hope to build is an array of the rectangular shafts that will be "pushed up", one at a time by a X/Y CNC machine from below. The goal is to be able to manipulate 10,368 shafts in just a few minutes.

To be honest, I need engineering help/advice to solve 3 major issues.
1) A way to rapidly push up each shaft to a variable position. I'm working on a solution in which a solenoid would energize in succession, essentially "pumping" each shaft to crank it up a notch each pump until the desired height is met. I'm working on a 3d printable model of a shaft that has a bottom plate that when upward force is applied it will "ratchet" the shaft up a notch; so energizing a solenoid under it 3 times would ratchet it up the 3 positions below baseline I need. That said, there's probably a better way and am open to any suggestions.
2) Speed. The 3d rendering of a terrain map should happen within a few minutes. I think this is achievable without having to have 1:1 solenoid to shaft as in the MIT's inForm machine, but I have no experience in making CNC machines fast... looking for any advice.
3) Cost. This is a hobby project. I don't anticipate this being a marketable product (at least in my application), and am doing it as a hobby. Cheap is king here.

Now, for the "why" of this machine. A couple years ago I built a table with an integrated smart TV to play table top games like Dungeons and Dragons. Since then I've often thought, how cool would it be to be able to represent a 3D dungeon (walls, passage ways, etc...) via robotic means. What I'm hoping to do is build another table with this machine built in (along with software to manage dungeon maps; did I mention I'm a software engineer?). So, like I said, its a hobby that will likely only ever go as far as my players, but think how cool it could be to spin up a 3D game board at will...

Any/all advice welcomed.