Well yes I think they do, although it depends on the situation itself but in my experience, in most instances where you find a person being empathetic towards a person or a cause then most likely they're going to be willing to cooperate with the people they're empathetic towards.
Of course there are cases of people who are empathetic towards a person or a cause but will still refuse to cooperate with people simply because they have their own personal reasons or agendas.
Well as far as I can tell, empathy begets cooperation but cooperation doesn't in any way beget empathy. There are people who cooperate with each other to achieve a common goal but not because they share each others feelings, simply because they both have goals that seem to be the same at that time.
Empathy on the other hand will almost always result in cooperation, as long as both parties don't have opposing goals. It's the absence or presence of that factor, "different goals" that determines whether or not a person will corporate with you, with or without any form of empathy on either side.
I hope this helps.