I took my son to a science museum where they had a gadget that many of us probably saw in movies involving a mad scientist. The gadget had two metal rods about two inches apart at the bottom. The rods were about six feet long, and four inches apart at the top. An electric spark would start at the bottom where the rods are about two inches apart. Then the spark would move up to the end where the rods are about four inches apart. The spark took about three seconds to get from one end to the other. After the spark got to the end, it would start again at the end where the rods are close together. I am fairly sure the spark is caused by a high voltage between the rods. What causes the spark to start at one end, and move to the other end?
The device you describe is called a Jacob's Ladder. You are correct that it is high voltage between the rods that produces the initial spark at the bottom of the ladder where the gap between the rods is the narrowest. Then the ionized air heats up, becoming less dense, so it rises. The current path rises as well because once a breakdown of the air has occurred, that ionized path is always the path of least resistance between the rods.
An easy way to test this explanation would be to place the rods horizontally instead of vertically. Presumably no movement along the rods from the narrow separation end to the wide separation end would be observed.