There are two kinds of libertarians. One type wants minimal government while acknowledging the nation-state as a sovereign entity (that needs to be defended by that minimal government). The other type reject the very idea of nationhood because they stupidly equate nationhood with "big government." The latter are committed to a world without borders (free trade and unrestricted immigration), and are hardly any better than the Left.
Unfortunately, most European "classic liberals" are libertarians of the latter variety - e.g. the op-ed writers for The Economist. Us libertarians are about a 50-50 split between the two, but the internationalist variety gets more visibility thanks to Reason etc.
The biggest U.S. Libertarian think tank and Libertarian ideas promoter, the CATO organization, is solidly in the open borders camp. I recently cancelled all my interaction with them.