Carrythroughs are just swingthroughs with an added pivot, or stance change. It’s that simple. Instead of a linear swing into the next trick, the smooth swing must be somewhat broken by a pivot. Typically, the stance change will occur first, followed by the swingthrough motion.
It is not uncommon for trickers to believe carrythroughs to be specific to frontswings, with or without a minor pivot, and truly the most common carrythroughs do utilize frontswings. The functionality of the frontswing vs the backswing swingthroughs should be recognized as identical, although reversed, meaning carrythroughs are not just simple frontswings. This understanding, that a carrythrough is a stance change with a swingthrough, opens greater possibilities without adding ambiguity or confusion with other tricks and transitions.
The carrythrough is often utilized in inverted tricks, and is covered more thoroughly in the Invert Landings sections.