One has a right to land if he performs the appropriate duties for that land, and only as long as he continues to perform them.This is a strange idea, today. What duties come with a house? We have a very different system, and it's not easy to even imagine most other systems. (One system we are familiar enough with to imagine is communism. But even that is hard. People vary wildly in how the imagine it, and all but one of the contradictory views must be mistaken. Some see it like We The Living, Anthem or 1984, and some imagine it as paradise, and others imagine everything in between.)
In our system of property rights, if you own a piece of land then it's yours, forever, to do whatever you want with. You can leave it sitting there, empty, or anything else. You have no duties. Except not exactly: you must pay property taxes. And you must abide by zoning laws and building codes. Properties can be declared "blighted". Properties can be confiscated if the Government wants to build a highway through them, or sometimes even if a stranger comes to your property and commits a crime there. And if you build a business you must abide by anti-discrimination laws, get a business license, and get permits to sell various things. If you wish to practice medicine on your property or sell alcohol, for example, that's heavily regulated.
So land owners do have duties, in a way. He must help pay for the upkeep of his cities. He must keep his property from being blighted. And he must follow various laws. If he does not do these things, he can potentially forfeit the right to his property. Still, we don't really think of these things as duties.
We also have an idea that the way to gain ownership of natural resources is to put them to good use. This doesn't come up a lot in America anymore only because so much stuff has already been claimed. Using a resource appropriate is a similar idea to duty. To say you can own an oil well but only if you use it by drilling it and selling the oil, and to say you can own an oil well only if you do your duty to drill it and sell the oil, is very similar.
Feudalism focussed on duties more. A lord gained certain privileges (ownership of land; right to tax his serfs; etc), but to maintain them he had certain duties (to protect his serfs in wars; to offer them justice in their disputes; to feed the poor; to help support the local church; etc). This system is not altogether without merit; both the lords and the serfs gain some benefit from the arrangement.
Emphasis on duty as I've described is a conservative attitude. That is not to say all conservatives would favor it or that it's their only option. But what I can say is that it's illiberal. Duties contradict the liberal ideals of maximizing freedom, tolerating diverse lifestyles (including ones that do not perform various duties), and an emphasis on people's rights.
I believe that the laissez faire style of capitalism, which eliminates duties, (thus going beyond the somewhat minimal duties we have today) is the best.
I do not believe the vast majority of libertarians, liberals, capitalists, or anyone else, could give a compelling argument against duty. You are welcome to try in the comments.