I don't have any hard evidence, but there may be a number of factors that influenced the sudden change in character.
The first time we really see Dumbledore as anything more than a kindly old man, and do anything more than general headmaster duties, is during the events of OOTP (Book 5). This was released in June 2003, around half a year after the Chamber of Secrets movie (November 2002), and a year prior to the release of the Prisoner of Azkaban movie (May 2004), which is when Michael Gambon assumed the role.
Before book 5, the novelized versions of Harry Potter contained only rumors of Dumbledore's prowess as a wizard, but had shown no real evidence for it. He was smart, experienced and accomplished, but other than that he was reasonably mysterious.
However, near the end of OOTP, when
Dumbledore battles Voldemort in the Ministry of Magic
this is the first evidence we have that Dumbledore is still a powerful wizard in his old age.
I don't know if the producers were aware of what would occur in the later books or whether JK Rowling kept it a secret, or if she even knew all of it herself at that point, but it's possible that Richard Harris played it in the way that he felt Dumbledore was portrayed in the earlier books.
However, once more was revealed about Dumbledore's character, they may have decided to start to change the character of Dumbledore to more powerful and active to reflect what further information they knew about him and establish him earlier so that it wasn't so much of a shock to audiences later on.
In addition to a new actor playing Dumbledore, the director of the Harry Potter franchise changed from Chris Columbus to Alfonso Cuaron between the 2nd and 3rd films. It may have been the directors choice to adapt the character to what he felt was more suited.
So the character change may have been the actor's decision, the director's decision, a producer's decision because of the new information about the character from the books, influence from the author due to her creative changes or a combination of all of the above, but it is likely one of these reasons. Had Richard Harris continued to play the character, he may have changed how he played it also.
Answer from Mike.C.Ford on Stack ExchangeYep, and I know that I'm going to receive heaps of backlash for this but Harris was simply better. People love to bring up the fact that Harris was suitable for the first 2 films and then Gambon took over at the right time, but if you take a look at Harris's career and how good of an actor he was, he would've done brilliant in the later films.
I think people see Harris as an old and frail man in real life and don't realize he was actually acting as one. And I think Harris would have shifted his character appropriately to suit the later films. Gambon always seemed out of touch with the spirit of the books for me. It was like he created his own character rather than making an adaptation of an existing character. His delivery of certain lines is kind of janky as well. The line "Happiness can be found even in the darkest of times" was clearly written for Harris and he would've delivered that and similar lines so much better. Harris had the perfect amount of whimsicalness, wiseness, intelligence and spirit.
I seriously think: people that think Gambon was better simply have not seen Harris's filmography outside of this franchise. He would've done so much better, and the franchise would have better off had he not died. I will die on this hill.