Dr. King’s most famous thought was that he wanted blacks to be judged by their merits, not the color of their skin. That’s the exact opposite of what BLM is pushing today. So yeah, Dr. King probably is turning over in his grave based on what the inheritors of his cause have done with it.
I also could not imagine Dr. King, a man of the cloth, would embrace the concept that one human life is worth more or less than another based on skin color.