I don't think some of the other comments here are entirely answering the question you asked, so I'll give what I've been told. Expenditure of calories in everyday life is for keeping things like your heart beating, lungs breathing, etc. After you have done everything to keep yourself living, the rest goes into work. If you were to do the same amount of work both running and walking, then the energy you put into it is by definition the same.

The problem arises when you take into account the fact that you need to take strides. When you take strides, it varies greatly depending on whether you are running or walking. When you are running, you tend to go higher with each stride and there is more motion going into it such as your arms moving more. Thus, there is more energy, measured in calories, spent running rather than walking.

TL;DR: If the motion between running and walking is identical and just sped up for running, then the amount of calories spent doesn't change. It's the change in technique that makes running spend more calories.