It's very difficult to pull this off, I think - there is a serious risk that the audience will be so distracted by the historical "wrongness" of (say) a black Mr. Darcy that they are constantly reminded they're watching a movie, a fake thing, and the suspension of disbelief is destroyed.
However, it is possible, at least where the film is not striving for great historical verisimilitude. Armando Iannucci's David Copperfield last year was joyously successful in casting with complete disregard of ethnicity - to the point that, if I recall correctly, at least one character had a parent of a totally different racial origin.
One could have imagined it working in a movie like The Favourite, too. And of course movies that are not based on actual historical figures or well-known fictional characters have fewer obstacles to overcome in terms of audience perception.