What eventually happens to the duke and king in the adventures of huckleberry Finn?

They are tarred, feathered, and run out of town on a rail (a log split into quarters, with the pointed, splintery side up to sit on- not the bark-covered round side). Very few people who are tarred, feathered, and run out of town on a rail, come back for more- which is the point. While it is not mentioned in the book, miscreants had their clothes removed, which made the hot tar even less pleasant.

Rails were used to make fences, so they were always handy. Abraham Lincoln had something of a name as being a good rail splitter.