Plumbing – Dishwasher drains with drain line detached, but not into the drain line itself

I have a Whirlpool dishwasher (model # W10632077A) that has been a problem from the day we moved into this apartment. It's never cleaned dishes properly, and now it's suddenly stopped draining.

The dishwasher drain line is connected to the sink drain through a disposal unit. There is also an air gap installed.

I went through the following steps, making sure to turn off the breaker when working under the dishwasher. After each step I tried to engage the drain pump to see if the dishwasher would drain, manually adding a gallon of water to the bottom of the tub with a jug where necessary:

• First I checked the filter and the drain pump inlets inside the
  dishwasher. There was debris, but no obvious blockage at any of the
  pump inlets/outlets that I could feel. I cleaned what I found and
  replaced everything.

• I then disconnected the drain line at the disposal. I did not find a
  clog in either the disposal end of the drain line or the inlet on the
  disposal itself. There is only a trickle of water in the drain line
  when disconnected from this side.

• I also checked the drain line connection at the drain pump under the
  dishwasher. No obvious clog at the outlet of the drain pump or the
  inlet of the drain line at this point, but no water inside the line
  either (the drain line on this side is transparent flexi-tubing with a soft rubber fitting at the end that goes around a ribbed hose adapter at the pump outlet).

When the drain line is detached water flows freely out of the drain pump with gravity. The impeller inside the pump seems to move freely when manipulated by hand through the drain outlet.

When you manually turn on the drain pump on the dishwasher there's no sound of water flowing, only a brief suction sound somewhere in the sink drain (maybe through the airgap?). It does sound like the pump motor is running, but it's not successfully draining anything. I didn't try to blow on the tube to see if the blockage was further inside; couldn't figure out a good way to do it without having to put my mouth on it (open to suggestions here, esp. if there's a purpose-made tool for this).

What I don't understand is why the water flows freely out of the pump when the drain line is detached, but does not flow into the drain line when it's connected. Could it be that there's a check valve between the pump outlet and the drain line that will only allow water to drain when under pressure, such as when the pump is engaged (and then maybe the issue is the pump after all)? Or does it have something else to do with the anti-siphonage design? I would expect at least some of the water to drain with gravity into the drain line and be visible if it drains freely with the drain line detached.