Why is dirty water coming out of the air gap when I run my dishwasher?
If this is only occurring when the dishwasher is running then your drain line on the down side of the airway is plugged or kinked since it just started happening. 

If the drain of the sink were plugged the water would be coming up from the sink drains as well. 

If the sewer line was plugged then you'd have other drains in the house backing up as well.

The airgap basically functions as a failsafe to help ensure water doesn't get backed up in to the dishwasher causing a flood. If it's spewing water all of a sudden when it didn't before, it's most likely the downside of the airgap is plugged.