Only one way to move new live gradients and that's by grabbing them by the center and moving them individually. You can't move multiple gradients at the same time.
If you have gradients in a folder and move the folder the gradients don't move with it. If you select multiple gradient layers and try to move, you cannot move them all at the same time either with the move tool or by grabbing the center of one of the selected gradients.
I was working with 10 graphic layers, each with their own gradient grouped with them. I had put all 20 layers in a folder but when I moved the folder the gradients didn't go anywhere. I had to move one at a time to get them in position again. A real pain. I ended up having to rasterize and merge everything to be able to move the group all at once, which totally defeats the purpose of the new live gradients in the first place.
You should be able to move multiple gradient layers at the same time without rasterizing either by selecting them all and using the move tool, or by putting them all into a folder and moving that.
Please add this functionality and the new live gradients will be a lot better.