The premise of the question is not entirely correct. The rise of the far right is a reaction to an aspect of globalization, but it is not "cross border trade, market fundamentalism, austerity, global institutions" that are fundamentally major problems for the far-right, but rather immigration.
For instance, Brexit was largely tied to fear of immigration. The U.S. ex-president Donald Trump launched his campaign by claiming that Mexican immigrants were bringing drugs and crime and promising to build a wall. French candidate Marine Le Pen promised to dramatically cut immigration and disparaged Muslims, a strategy that caused French president Emmanuel Macron to adopt certain elements of her position. Even outside of the United States and Europe, politicians such as Jair Bolsonaro have taken actions suggesting that they see immigration as a threat.
This is because the far right is founded on notions of a pure national and cultural identity, with undertones or overtones of explicit racism. The far-right generally rejects international institutions because it holds that they are promoting immigration and values that will lead to the destruction of the traditionalist ethnostates that it promotes, not because it fundamentally opposes international partnerships. Witness Donald Trump scrapping NAFTA and replacing it with his own, similar free trade agreement.
Remember, neoliberalism, like most political philosophies, is not necessarily the most natural or only natural constellation of beliefs, so it should not be surprising that the far right overlaps with it in some respects but not in others. Similarly, many classical Marxist socialists want universal healthcare along with nationalization of all industry, while many social democrats want universal healthcare without much industrial nationalization; monarchists often want religion in government in conjunction with a single hereditary ruler, whereas evangelical conservatives generally want religion in government without a hereditary ruler.