Very often, on various SE websites, I see new users posting questions that are off-topic. This is a common problem, and I think our communication lines to help these users are usually quite effective.
However, one thing I do not understand is the fact that these questions are always greeted with several downvotes. This question on music.SE is a good example: while the question is not truly great, the only real reason why people downvote it is the fact that it is not suitable for the Stackexchange network.
Now, as per the Should an off-topic question always be downvoted? discussion on meta.SE, I think the consensus is that we should not use downvotes this way. Users will get the wrong message from this, since downvotes actually have a different meaning than close-votes.
Hence my question: Is there a way to discourage people to downvote a question when a close-vote would have been sufficient? And if so, would it be useful to implement something like that, or would there be too heavy downsides to it?