The 3 port valve in normal position lets air into the front of the bolt.
It moves back, seals, the air leaks past the bolt and fills the chamber.
When the trigger is pressed, the 3port valve, blocks off the air inlet and dumps the air in front of the bolt.
The pressure in the chamber forces the bolt forward loading a paintball, and eventually goes down the barrel.
When the trigger is released, the exhaust port is closed off and the air inlet is connected back up to the front of the bolt.
I'll do a new drawing, just too many mistakes in the first one.
Edit: ok here it is.

Did the same mistake...The launcher has just fired, the trigger is released and new air is on its way in. So the 3port push valve and the slide valve are both in initial position.
I've added a QEV on this drawing, it works just as the ones in paintball guns work.
The chamber is a little bigger and the bolt no longers moves all the way back, this is far from being an essential part in the design, it was just to show that the chamber volume could be made larger.
I added a few colours

, Dark blue is the bolt, red is the barrel, light blue is the moving part of the 3-port valve, green is the QEV piston.
The thing on the side is a (bad drawing of a) slide valve, when the red sleeve is pulled back, air inlet is blocked off and top part of the pipe is vented.
Help this hopes... can't believe i actually wrote that
