To run upwards on an escalator going the opposite direction. In other words, running upwards against an escalator headed down. Typically, dropboxing is done on extended escalators such as those found in many large US cities, such as the ones providing access to the Washington, D.C. metro rail.
Jim and Tony were really sore after a late night out ended with the two dropboxing the Chinatown metro to impress Kathy and Diane.
A prank involving the placing an image of a cartoon turd inside of somebody's dropbox.
People keep dropboxing in my public folder!