There's another way to look at this question. People tend to think in terms of supercontinents forming and then breaking up again due to convection currents in the mantle, hot material rising and causing rifts in weaker spots, possibly in old sutures where the continents were shoved together—but what is really happening is that ocean basins are opening and closing, and the ocean has an active role in subduction.
The opening and closing of an ocean basin is called a Wilson Cycle. It begins when hot material rising from the mantle stretches the overlying crust. As molten material rises, a rift is formed. The rift is widened as material continues to squeeze into it. If that rifting goes on long enough, through a broad enough swath of a continent, ocean water will eventually flow into it, and an ocean basin begins to form. The upwelling of hot material will continue to rise through that thinner area of crust, pushing the plates apart. The Atlantic Ocean is an example of a basin that is well along in the Wilson Cycle; eventually subduction is going to begin at its margins, and the whole shebang will pivot.
This will happen because at the edge of continents, sediments accumulate. The weight of those sediments, combined with the weight of the water, drives the heavier, denser edge of the oceanic plate under the continental crust, which is fatter and lighter. Eventually subduction begins, and the basin begins to close again. The Pacific Ocean is an example of a basin that's closing.
If you look at a map of the oceanic rift zones, you'll notice that the one in the Atlantic is pretty much in the middle of that ocean, but the Pacific rift zone has been pulled all the way over to North America above Central America. Subduction is actively occurring on all margins of that plate.
The simple picture is that the continents are moving toward each other across the Pacific Ocean while the Atlantic Basin continues to widen. The truth is more complicated. When plates subduct, the water in the crust lowers the melting point of those rocks, so partial melting occurs. The partially melted material begins to rise through the overlying rocks, because it's less dense, and decompression melting occurs. Eventually, the upwelling of hot material forms plutons and volcanoes above the subduction zones. Fore-arc and Back-arc [PDF] basins can form. As the oceanic crust is pulled under the continental plate, island chains and other chunky bits get sutured to the edge of the continent along with sediments, making it larger. Our world is ~4.6 billion years old, so our continents are really large, now. They're unlikely to rift through the ancient cratons that formed their hearts.
What will happen if subduction begins on the eastern side of North America before the Pacific Basin closes? The margin next to California is a transform fault; it's not subducting. Will it eventually push itself under that part of North America again, or will the transform zone get bigger? The hot spot that was driving the ancient Farallon Plate under North America was eventually overridden by the southwestern states (Arizona, New Mexico, etc.) forming a rift zone. Will it continue to rift or poop out?
There are computer models predicting what supercontinent may form next. They will continue to change as our understanding of tectonic processes gets more accurate.
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