Essentiallly, the interstage pressure in a recip will adjust itself to keep the mass flow through the upstream cylinder equal to mass flow through the downstream cylinder.
Let's say you have a recip running along and you raise the suction pressure. The LP cylinder moves a certain volume and because you have raised the inlet pressure, the mass flow increases due to the increased density. However, the HP cylinder is initially moving the previous volume/gas density which is less than what the LP cylinder is now moving. To increase the mass flow through the second cylinder, the interstage pressure increases until the mass moved by the LP equals the mass moved by the HP cylinder. You can go through the same thought process for a decrease in suction pressure, temperature, etc.
Now, that's a very simplied explanation since as you change pressure to a cylinder the temperature out of the cylinder, gas compressibilities, etc change. Also, cylinders have a volumetric efficiency that I've neglected but the general response is as I've described it. Similarly, if you have a bad valve in say the LP cylinder, the interstage pressure will be lower than it should otherwise be as the LP cylinder isn't moving as much gas as design. Therefore, the suction pressure to the HP cylinder decreases since that reduces the mass flow through the HP cylinder. There's no controls that is doing this, it's essentially just an application of the old law of conservation of mass.
Hope I haven't confused everyone ;-)