Sorry, a bit of threadcreep here.

The origin of Running Down (as a shanty) is a good point. I'm not an expert on shanties or collections but I can't remember seeing it published either. I assumed it was a English shanty because the version I heard (from a Cornish group) had the lines "Running down to Cuba for a load of sugar/ Make her run you lime-juice squeezers." English sailors were "limeys" and Bristol used to do a lot of trade in the Caribbean for sugar and tobacco. But on second thoughts, maybe limeys wouldn't call themselves "lime-juice squeezers." Overall, the words seemed just a little bit too composed to be really genuine, but I accepted it at face-value.

One website claims it as a Newfoundland/ Labrador song, and says it was a protest shanty because the crew would stamp their feet a lot when singing it if they thought they were being worked too hard. But I don't quite get it. There's nothing in the text to suggest discontent like, for instance, "Leave her Johnny".

I think you could very well be right in suspecting that it's been expertly cooked up from a number of handy ingredients.