The previous director didn't want to do a third film:
When director Chris Columbus decided he didn't want to direct a third "Harry Potter" film in a row, Warner Brothers was faced with the task of finding a new director to take on the franchise.
They eventually turned to Mexican-born director Alfonso Cuaron, best known for helming the independent film "Y Tu Mama Tambien," but whose credits also include "Great Expectations" and "A Little Princess," a favorite film of "Harry Potter" authoress J.K. Rowling. However, before attaching himself to "Harry Potter and the Prisoner of Azkaban," Cuaron says he knew little about the best-selling children's book series.
"I was completely ignorant about the `Harry Potter' universe - I mean, I knew it existed but I didn't know what it was about, I hadn't read the books or seen the films," Cuaron admits. "I wasn't even curious."
All that changed when producer David Heyman sent him the script and said he was interested in Cuaron directing the film. "I thought I'd read 20 pages and tell him that it's not for me, and I just got hooked on the script," the director laughs. He then asked Heyman if he could read the book before giving his answer. "The moment I read the book, I said, `I have to do this.'"
https://en.wikipedia.org/wiki/Harry_Potter_and_the_Prisoner_of_Azkaban_(film)#Production
