Box you might help to pack thus missing flight? (4)

I haven't cracked this one yet — but the brilliant team at Fifteensquared have:

A cryptic defintion. Filbert has built up a reputation of some devious definitions but we both feel that this one doesn’t work. For a start, we would never refer to a lift as a ‘box’ and neither of us would think of ‘packing’ a lift with people. The idea obviously that if you use a lift, you wouldn’t be using a ‘flight’ of stairs.