I would use "Edge/Vertex slide". This allows you to move along an existing edge. The shortcut is: double tap G. In this option, you will find that you can "slide" or move along an existing edge, but can not go beyond the edge you are on. So you can then type Alt to extend, or type C to keep the option enabled. This will allow you to move beyond the selected edge. Once you are able to place a vertex along the continuation of an edge, you can then use it as a reference to move or copy the geometry you wanted to move there.

I would use Edge Slide or Vertex Slide. 

This allows you to move along an existing edge. The shortcut is double tap G. In this option, you will find that you can "slide" or move along an existing edge, but can not go beyond the edge you are on. So you can then press Alt to extend, or type C to keep the option enabled. 

This will allow you to move beyond the selected edge. Once you are able to place a vertex along the continuation of an edge, you can then use it as a reference to move or copy the geometry you wanted to move there.