Can air travel cause packaging air pads to burst or deflate?
Simple answer: yes, air travel can deflate or pop air pillows. 

Longer answer: Cargo holds are at the same pressure as the rest of the airplane, it's part of the pressure vessel. Generally airplanes are pressured to about 8000ft, so the air is a lot thinner and therefore there will be a higher pressure differential between the air in the pads and the air in the hold. Thin plastic is somewhat porous and will leak some air, the rate this happens will be faster when there's a higher pressure differential. This differential is known to pop these air pillows to the point many will not use them when shipping by air. Styrofoam, cellulose pops, or crumpled paper are the preferred method. 

You can see these forces at work next time you fly yourself by taking an empty disposable plastic water bottle on board. Seal it with the cap at airport altitude before departure. After takeoff as the airplane climbs the plastic will bulge as the cabin pressure drops, and when you uncap it you will probably hear a hiss. Seal it back up and then during descent the bottle will crumple as the air pressure increases.