In Yalkut Yosef OC 101:10 rules that a Sephardi who is davening in an Ashkenazi minyan should not daven Nusach Ashkenaz as shaliach tzibbur. He can only be shaliach tzibbur if the minyan agrees to let him daven in Nusach Edot HaMizrach, and if not he should refuse the honor.
Among the reasons for this is a very strong preference (for kabbalistic reasons) that Sepharadim should daven in their own nusach, and this overrides being the shaliach tzibbur. But aside from that it would seem that the shaliach tzibbur in any minyan should be proficient in the nusach of that minyan, and therefore "cross-davening" should be problematic even between Nusach Ashkenaz and Nusach Sephard.
Why are Ashkenazim meikil to allow the someone who usually davens Nusach Ashkenaz to be shaliach tzibbur in a Nusach Sephard congregation and vice-versa? And why are they meikil to allow one to daven in a nusach that's not their own in the first place?