I need to push an edge in but by doing so the connected edges will also get dragged in. To do this with faces I can just extrude inwards but with an edge it just creates a new edge disconnected from the surrounding faces.
Here is the edge:
How can I duplicate the vertices on the top and bottom of the edge so I can push it in without moving the edges above or below it?
I don't know if there is a one step solution, but aside from just manually doing everything I think the new extend edge tool (⎇ AltV) is about the closest to a one-step as anything I can think of:
Note that the selection is extended towards the initial position of the mouse when ⎇ AltV is pressed, which can be confusing before you get used to it.
That said, I think in your particular situation I would do something more like this:
