There is a real difficulty with backwards compatibility between LR3 and LR4, and that is that Adobe introduced Process Version 2012 with LR4. PV2012 uses a completely different processing engine by default than LR3, and PV2012 is one of the big reasons to upgrade. You can though choose to use the older PV2010 (actually you still can, even in the latest LR CC version). Moving edits from newer to older versions of LR is possible, you will need to perform the edits in the normal way, and then ensure that you have LR write the edits to an XMP file. You will then need to import the images to the older version of LR, during which LR should read the XMP file, for images already in the catalogue you will need to have LR read the XMP file again. As long as you are using the same PV then any instructions that are not included in the older version are simply ignored. This could still be problematical, since there are significant differences between LR4/5/6/CC regarding the tools like local brushes, filters, spot healing, etc. all of which would be lost in the older version. To go back to LR3 you would need to really hobble the later versions of LR in that you would need to be using PV2010.
Alan