Generally, you assume the pressure in each tank is uniform.
If you're asking how quickly air or a fluid will go from one tank to another, there are equations for determining fluid flow through valves and piping. The Crane paper #410 is the most widely used reference for those equations.
The rate of flow is proportional to the pressure difference across the valve and goes inversely with the impedance of the valve, which is a function of the valve length, opening size and Reynold's number of the flow (which in turn, is a function of flow velocity, and is hence determined iteratively for a general case).
Hey P, if I were going to do a really quick and dirty analysis for something like that, I'd do a spreadsheet that calculated flow through the valve only and just oversize the pipe. The valve is generally your largest restriction by far, unless you have a ball valve. If using a ball valve, just reduce it a bit. Or go through the entire system and calculate the flow at a single pressure differential, and come up with an "equivalent Cv". The Cv is the flow coefficient for the valve. If you come up with an equivalent one, it makes all the math much easier since you don't then have to recalculate flow through the pipe.
Use the equations on this page for valve flow:
Then start a spread sheet. Make tiny steps, 0.0001 seconds at a time or something. First column in your spreadsheet is time, then pressure in supply, then pressure in cannon, then volume of cannon, flow of air through valve, etc... Each row gives you the conditions at a given point in time, and each column calculates one of those parameters.
You really need to incorporate the first law into it, because the temperature of the supply and cannon is going to change fast. If that's too difficult for you, assume its isothermal. At least you'll get some very rough estimate of what will happen.