Some countries will, some do already. The big limiter is transport costs. While many countries can produce corn cheaper than the U.S., not much of it can remain cheaper by the time it’s got there.
Brazil is the big exception. With a mature fuel ethanol industry, plenty of suitable agricultural land, government backing, proximity to America and a major import tax break from the U.S. for certain amounts of ethanol, it can and does send quite a lot north.
That will certainly continue for a while. The amount qualifying for the tax break is likely to remain cheaper than cellulosic ethanol even for many years after it becomes commercially significant as the conditions in Brazil make sugar cane (which it uses to produce ethanol) signficantly cheaper to grow than anywhere else within spitting distance of the U.S.
There is not likely to be much other trade of ethanol to the U.S., but some other signficant ethanol trade routes could open up round the world in the coming decades.
It’s worth noting though that, in all likelihood, no major country except Brazil is likely to be able to satisfy more than around 10% of its fuel needs with bioethanol (or any other biofuel) for at least a decade and I’d be surprised if many topped 20% before we’re sure when the oil’s going to run out.
Big plus: Replacing 10% of fossil fuel use with a biofuel whose growth and combustion involves fairly low net carbon emissions and that has the potential to, in a reasonable amount of time, be produced sustainably is a good thing.
Big minus: Some crazy people would much rather have food than fuel …
Resolution: Cellulosic ethanol processes (turning any cellulose, e.g. grass, wood chipping, tiger lillies, into ethanol) will soon be/are viable and will be commercialised to convert less versatile biological fuels into ethanol and will, eventually, make the convertion of corn, sugar beet, cassava and a variety of other tasty produce into ethanol, uneconomical.
Give it a decade