Cold air from gap underneath kitchen cabinets
Usually cabinet makers don't try to seal the gap between the toekick and the cabinet box.  What's most likely happening is there is an opening in your drywall behind the cabinets for plumbing or electrical that the installers did not seal, and that is your source of cold air.   

Since getting to that is near impossible (without dismantling the cabinets) your're going to have to settle for plugging the gap.  Maybe some fiberglass cut to fit, and some plastic tape to seal.  The space under that cabinets may run the whole length of the assembly.  It will be a kludge no matter what.  Just make it look semi-reasonable.

That register; is it a return or a supply?   A return may just come up in the floor, and they cut the toekick to provide flow.  A supply should be ducted to the register cover.