I know Les Miserables is set in 19th century France which was obviously dominated by whites. However, it's one of those productions where the race of a character does not really matter- it does not change the plot. There are some productions where you cannot change the race of the character whatsoever such as Hairspray because racism is crucial to the plot so it would not make sense to have a black Tracy or a white Seeweed would it? However even though Les Mis is set in a time where there were little to no non-Caucasians in France, changing the ethnicity of the character does not really affect the plot because race is not essential to the plot. If it were, then yes. Don't change the race of the character. But since it is not, I see no problem in it. This does not just go to black people. It also goes for Asians, Arabs, Hispanics, etc. Heck I wouldn't mind a white person in Lion King because like with Les Mis, although Africa is dominated by blacks, race does not affect the plot and besides, they will be playing animals so people probably won't even acknowledge their faces that much.
In conclusion, I think it is okay to cast non-Caucasian people in Les Miserables because their ethnicity does not affect the plot