Hispanic actors broke the barrier in Hollywood a while ago - think Dolores del Rio and later, Anthony Quinn. Even before them, Lita Grey, Chaplin's second young wife, on her mother's side was from Californian Spanish/Hispanic families. So it seems that the biggest barrier would be the accent.
With African-Americans, the situation is was more complicated. To remember Hattie McDaniel, the first African-American Oscar winner.
Having received the Oscar,
"McDaniel then was escorted, not to the Gone With the Wind table — where Selznick sat with de Havilland and his two Oscar-nominated leads, Vivien Leigh and Clark Gable — but to a small table set against a far wall, where she took a seat with her escort, F.P. Yober, and her white agent, William Meiklejohn. With the hotel's strict no-blacks policy, Selznick had to call in a special favor just to have McDaniel allowed into the building".
I think today the choice of the actors depends on the material. Maybe as our country becomes more diverse, we shall see Asian, Indian, African-American and Latino actors in larger amounts.