The libertarian position, as I see it, entails no possible position on "immigration," because immigration is a state-based concept. If there is no state, there can be no immigration, only traveling people. In a world with states, this traps the libertarian into taking a position on a fundamentally statist idea. Open borders, managed borders, and closed borders are all statist concepts.
The question ought not be whether the borders should be open, and I think no libertarian should take a position on whether they ought. Rather, the libertarian should recognize that if all property were private, then immigration policy would not exist as a concept at all.
"Refugees" would only come to such a place if they could survive there on their own, or if they had a sponsor supporting them. In either case, the likelihood of extremist behavior from "immigrants" would be drastically reduced due to either their self sufficiency or the increased scrutiny for which a sponsor would be held liable. Today the State as sponsor is held liable for nothing.
Additionally, since security would all be provided privately, there would be more security with greater firepower in closer proximity to any events that did occur. Private security is highly visible because its purpose is deterrence, and in an unhampered market would have the tools to handle whatever is necessary due to being held liable for failure.
However, there does not seem to be any possible peaceful solution, given the current social conditions of universal statism.
Ludwig von Mises points out that in a society under the conditions of democracy and interventionist ideology, there exists an endless civil war between cultures and nationalities inside of a cosmopolitan state. While in democracies this war is conducted through political means for peaceful power transfers, the conflict still exists as each special interest or nationality attempts to gain control of the state. This is not only to protect itself from other nationalities, but to impose their views on others. It is a natural instinct of man to desire maintenance of his way of life and to resist change.
Mises' analysis is purely positive, meaning he does not offer any normative arguments, nor does he offer any policy prescriptions. He merely analyzed the social conditions in a society where there exist many different cultures and nationalities, and pointed out that there will be problems with such a society under interventionist and welfarist ideologies.
He also points out this antagonism would exist to some extent in a purely laissez faire social system. It appears, at least to me, to be an unavoidable and irreconcilable conflict, interior to social units. Therefore, social units composed of voluntary arrangements will result in the most possible people living under their desired "policies."
This does not help us, today, however. Under the current conditions of statism, the likely best solution to immigration problems is ending the welfare state, and ending international wars. Immigration, illegal, legal, and refuge seeking are symptomatic of the disease.