Yes, they do lean and fall down, and the vine runs on the ground. If the wind blows them around they look funny for a while, but the leaves will turn up towards the sun again. The fruit is usually on the ground because it gets so heavy. They keep putting on fruit till frost here.
Gardening at 5000 feet elevation, zone 4/5 Northern Utah, Frost free from May 25 to September 8 +/-