There is an interesting shift in the paradigm of casting between Twilight and The Twilight Saga: New Moon and, newly, The Twilight Saga: Eclipse.
In Twilight, most of the cast members chosen were from the United States, including actors Kristen Stewart, Taylor Lautner, Ashley Greene, Peter Facinelli (dual citizenship with Italy), Elizabeth Reaser, Kellan Lutz, Nikki Reed, Cam Gigandet, Anna Kendrick, Michael Welch, Christian Serratos, Gil Birmingham, Justin Chon, and Billy Burke. Robert Pattinson is from the United Kingdom, Rachelle Lefevre is from Canada, Jackson Rathbone is from Singapore (though he grew up all over the world, including in the U.S.), and Edi Gathegi is from Kenya (educated in the U.S.).
In New Moon, the additional cast members were of more various descent. While there were some American-born actors added to the cast (including Alex Meraz and Dakota Fanning), many of the newest cast mates were from Canada and the United Kingdom. From Canada for The Twilight Saga: New Moon are Christopher Heyerdahl, Daniel Cudmore, Graham Greene, Noot Seear, and Cameron Bright. From the United Kingdom are Michael Sheen, Jamie Campbell Bower, and Charlie Bewley.
For The Twilight Saga: Eclipse, the newest cast member announced is Xavier Samuel, an Australian.
The global nature of interest in the Twilight series has certainly opened the casting process to a wider sphere of interest.
While Twilight had a very stream-lined cast, with the exception of Pattinson and Lefevre, of mostly those from the United States, New Moon has notably been more international in nature. Since New Moon was filmed in Canada, it makes perfect sense that many of its additional cast members would be from Canada. Still, it is interesting to watch the diversity of nationality unfold as casting for the films commences. With Eclipse, there are still two major cast members yet to be announced, including Seth and Leah Clearwater.
With The Twilight Saga: Breaking Dawn, the nationality of cast mates will be at its pinnacle of diversity (as necessitated by the book's description of the newest characters' origins).
So, other than convenience, is there any other reason why the casting has become so diverse?