To move the active layers (including linked layers) or selection using this tool, in the Texture view click and hold down the mouse button and drag the layer/selection to the desired position. Parts you drag outside the texture area will be cut off.
In the Attributes Manager you will find an Auto Select Layer option. If this option is disabled, the active layer will be moved. Otherwise BodyPaint 3D will choose which layer is moved based on which material channel is active, which layer has paint on it, and where the mouse pointer is when you click at the start of the move layer action.
The following move layer modes are available: