You can copy and paste a Layer and all its content from one Graph to another (adding a new Layer) by just selecting the layer and using Ctrl+C to copy and selecting the new Graph and Ctrl+V to paste.
For copying just content you can use script:
win -a Graph1; // Make your source graph active page.active = 2; // If multiple layers, make source layer active layer -c; // Puts count of plots in 'count' and names in '%Z' win -a Graph2; Make your target graph active page.active = 3; // If multiple layers, make target layer active layer -i %Z; // Includes all datasets using existing plot styles or layer-i201 %Z; // Includes all datasets as Scatter plots
The key is to make sure you have selected the source layer when you do the copy. You will know you have selected it when you see the rectangle and 8 selection handles (in 4 corners and 4 edges) around your graph. In case you have trouble getting the selection (e.g. your data is very dense or you have a contour graph) try holding down the Alt key and then clicking on the graph.
Once you have the selection, press Ctrl+C. In the new graph window, click where you want the upper left of the copied layer to appear and press Ctrl+V.