Zevahim 15: When Is Conveying Not Walking?

More on whether the conveyance of the blood from the place of slaughter to the altar needs to be via walking. Plus, how to handle a bird, whose slaughtering is already at the altar, without collection and conveyance of the blood. The discussion, when reported in the land of Israel generated laughter, which is eventually explained. What about a non-kohen doing the conveying? With all kinds of possible permutations.