I remember that other famous MLK quote, "The only way the white man will show empathy and solidarity with the black man is by making him miss his flight." Actually, I imagine MLK knew what would be counter productive and what wouldn't which is why he never blocked access to New York International Airport.
It's not about empathy or solidarity. That's why it's called disruptive protest. And Slayven already mentioned in this very thread that if this is what causes them to go "I can't support their fight against oppression anymore because it inconvenienced me" then those people were never going to do anything meaningful in the first place.