Doesn't the International Date Line sweep around the Earth in actuality?

I mean they always show it on maps as this neato looking zig zagging line that splits Alaska and Russia and dodges around various islands in the Pacific and goes wide for New Zealand's sake, but in reality doesn't the IDL sweep across the earth along as the planet spins? I have an easier time picturing a 24 hour gradient of time from any one spot on the planet circling around back to the point of reference, but they never describe it like that. If somebody travels a little bit before or after midnight east and west far (and fast) enough wouldn't they be hopping back and forth between the last day and the current day for a while, regardless of where it was they did this? Other than for easy visuals it makes it seem if you went east/west or west/east just so over the IDL in the Pacific you would gain or lose a day repeatedly, which makes no sense at all. Common sense would make me think the IDL moves across the world along with the day and night however.