It's most likely due to an increase in the rate and depth of your respiratory rate; chugging in lots of dry air has an irritant effect on your upper respiratory tract, triggering your cough reflex.
As Obi says, the only sensible thing I can think of to combat it is making sure you are well hydrated.
Nasal strips would probably make it worse; by artificially opening up your nose, you would reduce the warming moistening effect of your turbinates; the folds which cause turbulence in the air you breathe in, which helps to reduce the irritation caused by dry air, and helps in gaseous exchange; warmed moistened air is niiiiice for your lungs…