Explain to me how "property rights" can be part of natural law in a world of limited resources into which more and more people are born. Are the latecomers just unlucky that there is no unclaimed land around? How much property can one person morally claim?

I’m no expert but imho one can only claim as much as one can properly manage. Most of the land in the world has been claimed by institutions and states, there is probably enough land for individuals.