Guild Wars 2

Let’s say that I have an infused backpiece and an uninfused backpiece. Do I need to select the skin or the stats of the infused backpiece to transmute the additional infusion slot, or will it disappear in the process and I would need to infuse the outcoming product?

You would select the stats of the infused back-piece, like any other transmutation.
The infused item is separate from the original version(un-infused version) hence the extra slot being part of the infused piece.