I had to search for a little while to find a topic "the returning cough"....but we talked about with Beckham that he does the same type thing and here is a post from that.
It sounds like a Reverse sneeze and that is an irritation of the soft palate and throat that causes spasm's. Sometimes the dog will hold it's neck extend and the chest will expand. Bacially scare you and make you think they are gagging, coughing or gasping for air.
What causes it....well.....Anything that bothers the throat can cause it. Being excited, exercise, eating, drinking, pulling on a leash to hard, pollen, perfumes, viruses, household chemicals, allergies, and the most common post-nasal drip.
Dogs that are considered "Brachycephalic" which Boxers are!!!! have a short snout and a elongated soft palates occasionally suck the palate into the back of the throat. If the palate is real long sometimes they have to have surgery. This is pretty common in Pugs.
If/When it stops, the spasm is over. You can massage the throat to make it stop. You can put your hand over your dogs nostrils, which makes them swallow and clear the throat. Some dogs have this their entire lives; some only when they are older. The spasm is only a temporary problem that goes away on its own...leaving no damage.
Normally with bronchitis they cough/gag on and off all the time. Even more at night or while resting. If there is something acutally caught in their throats they will paw at the mouth and throat with their front feet almost constant.....
So I am thinking maybe the drying of the mouth/throat and sinus is causing a reverse sneeze.
Search topic The returning cough and see if it sounds similiar to what Beckham was doing.....Good luck!