YOur media is what you have used as a platform for your beneficial bacteria to colonate. It may be ceramic cylindars, sponges, lava rock, bio balls, etc..... what ever it is, it should be transported in the same container as the fish. Their waster (what little there is) will sustain the colony of beneficial bacteria during the move - and will help keep the water clean for the duration, too. By bubbling the air up through it, you will be moving the water past it and adding the oxygen the bacteria need to process the waste.
I usually bag it up in old stockings or bags that you can sew or buy from the fish store. Drop these bags into the fish containers and all is good. You can fish them out easily and put them back in your filter when you arrive at your new home, and, though your cycle may bump a bit, it should be intact enough to bounce back quickly enough with a bit of care.
