This weekend Texas senator Ted Cruz spoke about how “we all face the same challenges, including a bold and global left, that seeks to tear down cherished national and religious institutions”. Nothing to see here, you might think – except that he was not addressing a local branch of the Republican party in Texas, or a conservative US media outlet. He was speaking on screen to an audience of thousands in Madrid, at a meeting of the Spanish far-right party Vox. It was one of many recent outreaches to the global far right by US rightwing figures, which seem to have increased since the ouster of Donald Trump.

Is the so-called “Populist International”, so often foretold but never realized, finally taking shape? And will the US conservative movement play a leading role in it? Or is this more about domestic politics than global domination?

Unsurprisingly, given that the US conservative movement, like the Republican Party, covers a broad range of different shades of often far-right ideology, different people have spoken to different types of far-right groups. There are at least four major strands of far-right international networks in which US “conservatives” of all levels participate.

The first and most important is the global Christian right. The US Christian right has long been a global player and has been particularly active in post-communist Europe – as is captured well in the Netflix series The Family. They have found influential supporters in Russian president Vladimir Putin and, more recently, Hungarian prime minister Viktor Orbán. It was at the latter’s invitation, at the bi-annual Budapest Demographic Summit in Budapest, that Mike Pence recently spoke, together with a broad variety of academics, church leaders and politicians from around the globe, including the French far-right maybe-presidential candidate Éric Zemmour.

Budapest has also been the new promised land for the second strand, the so-called “national conservativism” movement – the brainchild of the Israeli think-tanker Yoram Hazony. National conservatism is a kind of far right for people who read, to put it dismissively – an attempt to merge the already ever-overlapping conservative and far-right ideologies and create a far-right movement fit for the cultural, economic and political elite. Tucker Carlson gave a keynote at a national conservatism summit in Washington DC in 2019 and recently took his Fox News show to Budapest, where he raved about Orbán and his regime. And the Conservative Political Action Conference (CPAC) is said to be hosting its 2022 meeting in Budapest too.

The third strand is the long-standing connections between some far-right Republicans and the usual suspects of the European far right, like the Austrian Freedom Party (FPÖ) or French National Rally (RN), which are built on a shared ideological core of nativism, authoritarianism and populism. Connections between the European far right and Republican members of Congress go back decades; think of people like Steve King, the Iowa Republican, and Dana Rohrabacher, the California Republican. They were fairly marginalized within the party – and both have, ironically, lost their seats in the Trump era. It was largely with these groups that Steve Bannon created “the Movement”, which never moved beyond media hype.

And, finally, we have the party that Cruz sent a supportive video message to, Vox in Spain. Almost completely under the radar, Vox has been building a conservative-far right network in the Spanish-speaking world, partly facilitated by the party’s Dineso foundation. Focused mostly on Latin America – and piggybacking on the Latin American right’s long-standing fight against “communism” and for conservative Christianity – the foundation has published a “Charter of Madrid” signed by more than 100 politicians and political activists from Europe and the Americas, including US conservative activist Daniel Pipes (anti-Islam) and Grover Norquist (anti-tax), as well as a host of Latin American MPs. The particular meeting Cruz spoke to was attended by various European far-right leaders, including Giorgia Meloni of Brothers of Italy (FdI), currently the biggest party in the polls, and André Ventura of Chega in Portugal.

Obviously, these international networks overlap on many issues, most notably in their common opposition to the “global left” but also, in different gradations, to immigration, Islam and “gender ideology”. But they also disagree on central issues, from the importance of religious doctrine to the role of Russia, and consequently have very different and shifting memberships. And they differ in the role of the US conservative movement within the network.

With the exception of the Christian right, which has long dominated the global movement, the US does not play a leading role in these networks. Even the “national conservatism” network is run by an Israeli and increasingly funded by Hungarians. Moreover, the various US Republicans who have recently participated in these meetings seem to use their international connections more for domestic gains – most notably in the fight for the Republican nomination (should Donald Trump not run) – than for the sake of building a Populist International.

This is not to say there is nothing new to recent developments. In the pre-Trump era, only relatively marginal rightwing conservatives and Republicans had open connections to the international far right. Today, the ties between the broader US conservative movement and the global far right have become mainstreamed, from the Republican party to National Review, with fewer and fewer dissenting voices. Still, steeped in US exceptionalism, the US conservative movement remains mostly inward-looking, using international connections and events primarily for national political struggles. And the Populist International is still more media hype than political reality.

• Cas Mudde is Stanley Wade Shelton UGAF professor of international affairs at the University of Georgia, the author of The Far Right Today (2019), and host of the podcast Radikaal. He is a Guardian US columnist