The issue seems to be in the Blend Control portion of the A/C control system. The vent you hear opening is actually the blend door. This is the door that opens / closes blending your heat from the heater core and your cool from your evaporator to bring your cab down to whatever you have the thermostat set for While the Control Head (or whatever Dodge uses for the climate control logic system) determines the blend door position, it uses a variety of sensors to determine position.
If it's a feedback system, it actually monitors the position of the blend door by monitoring the voltage of positional sensors inside the motor or actuator. This movement back and forth indicates either a failure of the circuit to it, an inaccurate input to the logic (possibly due to a poor connection as well), or the Control Head itself. You can pretty much rule out a mechanical failure, like a broken door, or bad motor just because of how it acts initially.
I'm not sure about the complexity of the system, but it sounds like a trip to the dealer is necessary, unless you have access to service information, and the diagnostic tools to properly troubleshoot this. Whatever the case, good luck!