That all but publicly happens here in Illinois from the district reps in Chicago or their little support team. They try to influence a lot of down state reps to vote their way. Or better yet the mayor of NYC jus recently donated over 2 million to get a representitive elected to a district in Chicago because he supported new gun laws. I'm not sure what the problem is with contacting representatives from other districts, especially in your own state, regsrdless if they represent you or not they are still a rep in the entire state and what they do or don't do effects the entire state.
