I agree with the idea of not over doing it.
But, I think this issue which I have had on a regular basis since 2008 has to do with scar tissue in the muscles.
So, it's the same muscles (quads) but used in different ways.
I think the muscles get scar tissue with a grain going one way for cycling, and the opposite way for walking down hill. Since I'm cycling 12 hours a week, every week, the transition to walking from cycling is worse.
Anyway, who knows? The main thing is that I figured out how to live with it and avoid the most likely scenarios for pain, and have still managed to do 7 long distance walks (>100 miles each) without too much pain.
I freely swing my legs for 20 minutes on the off days thinking that breaks down the scar tissue more gradually.
But again, who really knows if this is voodoo or not.