As a member of one of the most underrepresented communities in American media in proportion to its share of the population, I don't really care at all. As for whether it can be done well, I feel like it depends on the medium.
On the stage? Absolutely. One of the best Julius Caesars I have seen on stage was that of a Black British actor with the RSC.
When it comes to film and television, it's a bit trickier. I don't think we should see Britons of South Asian or West African ancestry in supposedly realistic period dramas set in the Tudor court. Or blonde Swedish actresses playing Lady Fujitsubo in a film adaption of Genji Monogatari. It just ruins the setting, which is also why the Memories of a Geisha adaptation is just painful to watch. Zhang Ziyi and Gong Li are both great actresses, but when it comes to the world of Gion and the whole Maiko culture, they're daytrippers. Just because most of us in the US can't tell the difference between the Chinese and the Japanese is not a good reason to cast Chinese actresses in what are quintessentially Japanese roles.
On the other hand, what is acting if not becoming someone else? Kenneth Branagh, Bill Nighy, Eddie Izzard, Terence Stamp, and Tom Wilkinson are all British thespians, but they were supremely believable as Nazi generals in Bryan Singer's Valkyrie. Phil LaMarr is an African American voice actor and yet he is absolutely perfect as Ramza - a noble in Ivalice which in the game is coded as an European kingdom.