It might be inutitive to assume that the Earth is fixed and everything is moving with respect to it, but that's just because the Earth is so big and heavy. Every motion you, a car, or a plane makes it ultimately pushes itself off of the earth, which seems to sit still.
When you consider other things, though, it becomes less clear which is "moving" and which is not. You see a plane flying fast overhead...but if you're on the plane, you seem to be sitting still. Indeed if you were knocked out and woke up on a plane, you wouldn't even know you're moving 600 mph unless someone told you. If you're driving a car, it doesn't really matter whether you get hit by an 18-wheeler or the 18 wheeler hits you. The effect is physically and mathematically identical.
Likewise with the "moving pavement". You are moving relative to it as much as it is moving relative to you. This might be hard to grasp because the Earth has appeared to be "fixed" all your life. Perhaps then you have to leave the Earth (at least in your mind) to overcome this. Imagine being out in space flying around. You might be moving 20,000 mph relative to earth, but relative to the sun you might be moving ~80,000 mph. Likewise the Earth is moving relative to the sun at 60,000 mph. Relative to the galaxy ~200,000 mph? What if you're an alien that came from another planet moving at yet another speed? A So which one is right? The short answer is that the're all right...it just depends on which frame of reference you look at. Usually we pick the one we're about to crash into.