It depends on the discipline ie. Skeet, Trap, Sporting.
For example for Trap all targets have to be shot swing thru as it's impossible to guess the direction of the target, thus a proper target line has to be established in order to break it.
On the other hand for Skeet maintained lead is popular as the path of the target is very regular, thus it is most efficient to start well in front of the target in the correct lead.
As far as Sporting, which includes from Five Stand, Compak to Sporting Clays/English Sporting and FITASC, there are a variety ways of breaking clays, depending on the presentation. For example the king of Sporting, George Digweed shoots swing thru most of the time. Other champions use a combination of methods depending on the target presentation. For quartering targets coming from behind (swing thru) is an effective method but for crossers pull away is a popular choice (see Ben Husthwaite Start Method) . There are some shooters such as Gebben Miles that use maintained lead on crossers. And there are others that use a form of move, mount, shoot such as world champion David Radulovich. Because of the complexity of target presentation and possibilities in breaking a target, it takes a lifetime of improvements to master this sport.