Swan/Dolphin transportation is provided by Disney ... thus it is the same as the YC/BC and the Boardwalk. In other words, it IS Disney transportation ... just like the other Disney hotels on property.
That said, we stayed at the Dolphin and while it was very nice ... just didn't have the same "Disney feel" as the YC/BC, AKL, POFQ and POR ... from our perspective. Can't put my finger on exactly why ... but if we have the monies we stay at the AKL, BC/YC before staying at the Dolphin even if we can get a better rate. There are many others that would disagree but that is our opinion ...and what we found for our family.
The transportation here, though, is the same as when we stay at other resorts in the area.