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It's basically orthodox leftism. Leftist governments were historically against immigration because of domestic labor laws. Liberals like to paint anything other than utter globalization "fascist".

And no, nazis didn't choose the "national socialism" because they believed in "socialism but national version", they were anti-communist:

Socialism’, he retorted, putting down his cup of tea, ‘is the science of dealing with the common weal [health or well-being]. Communism is not Socialism. Marxism is not Socialism. The Marxians have stolen the term and confused its meaning. I shall take Socialism away from the Socialists.

‘Socialism is an ancient Aryan, Germanic institution. Our German ancestors held certain lands in common. They cultivated the idea of the common weal. Marxism has no right to disguise itself as socialism. Socialism, unlike Marxism, does not repudiate private property. Unlike Marxism, it involves no negation of personality and, unlike Marxism, it is patriotic.

‘We might have called ourselves the Liberal Party. We chose to call ourselves the National Socialists. We are not internationalists. Our Socialism is national. We demand the fulfilment of the just claims of the productive classes by the State on the basis of race solidarity. To us, State and race are one…

So basically they grifted on the name "socialism" to appease to workers, and redefined it as "national capitalism based on race".