] Poor old Emma is making all the running.She is trying to overcome her dislike of Jane and trying to draw her out.

This IMO is not a good example because Emma here is only trying to draw Jane out to satisfy her own curiosity about Frank Churchill, and taking no cues from Jane. But a perfect example is when Elinor is left to perform the duty of making conversation with Mrs. Jennings and all their acquaintance because Marianne is both too wrapped up in herself and too disdainful.