I think it would depend on the country, the crime, the circumstances, the embassy involved, and the amount of political pull you, or your lawyer, has.
In Cuba, since Canada sends a lot of tourists their way, the Canadian Embassy has some major pull, and I'm sure they are willing to use it. Of course, for the US, the opposite is true in Cuba. The US diplomatic mission in Cuba has zero pull, and unless you work for the CIA, there's not much they can or will do for you there.
In the Dominican Republic, while the US Embassy has major pull, they are hesitant to use it in criminal cases that aren't clearly falsified, or that involve a person of some notoriety. Their pull is used mostly for commercial interests, and they are not really interested in expending political capital on criminal cases, aside from ensuring basic representation, and perhaps notifying your family.
I think the best course of action in the DR is to basically stay out of trouble, and don't plan on a whole lot of help from the boys at the US embassy. If it is clearly a falsified case, you might get some relief, but you had better be absolutely certain you are able to prove what you claim, to both the DR authorities and to the US Embassy. Otherwise, I imagine you'd be in for some tough sledding.
This is just my opinion, and you should not confuse anyone's opinion for established fact. Just be careful, have access to a good lawyer, and don't expect to be facing a fair or particularly honorable justice system. Remember, money talks, and things aren't always as they seem.