The dopesheet behaves similiar to the 3D view.
The right mouse is used for selecting objects or keys. Incidentically, the RMB followed by a movement before releasing the mousebutton selects an object/key and immediately enters the translation operation.
But since you only want to do a translation, you have to initiate it with a shortcut as intended. The correct way is to do a box selection and then...
- make sure the cursor is hovering in the dopesheet (so it will receive the keys pressed and not some other window; Blender is context sensitive)
- press G (grab) to enter a translation operation (G key down and up)
- move the mouse, (moving the keys)
- press the left mouse button to confirm the operation and exit the translation.
4a. press the right mouse button to escape the operation. The keys will snap back into place.