While the British Brexit debate rages on, it continues to ignore entirely a more important European political battle: the search for the best way to defeat populists on the continent within the next eight months – the time left before the EU parliamentary elections. That little of this seems to get factored into internal British discourse is not surprising: for all the headlines about Theresa May’s “Salzburg humiliation” or “EU dirty rats”, Brexit is essentially the British talking to themselves.

Across the Channel, a new line of attack against Europe’s populists is taking shape: it focuses on breaches to democratic rule of law, rather than the issue of immigration. That’s why the most important piece of EU news this month was not the Salzburg situation (entirely predictable) but the 12 September vote in the EU parliament on the rule of law in Hungary (much less so). For the first time, an EU institution which is hard to describe as “anti-democratic” (it is elected directly by its citizens) called for the activation of article 7 procedure against a member state’s government because of the way it has been disemboweling essential democratic institutions and rights.

For a long time now, Europe’s liberal democrats have been struggling to curtail political forces that threaten core principles. But since the 2015 refugee crisis they have let themselves get dragged into precisely the debate that populists can thrive on: migration. Not only was the EU at a loss over how to deal with the arrival of a million people in 2015, but its liberals have mostly failed to convince large swaths of the population that immigration is needed, that it needn’t upend social services, and that it does not spell the end of a certain sense of European or national identity.

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Migration conjures up fears that rational argument struggles to cope with. Hungary’s avowedly “illiberal” Viktor Orbán and Italy’s far-right Matteo Salvini have secured major electoral breakthroughs by relentlessly pounding away at migration, depicted as a “Muslim invasion” (Orbán) or as something that requires “mass cleansing, street by street” (Salvini). With that rhetoric, they are now preparing to launch their bid to take control of the EU parliament, along with like-minded European politicians.

With that rhetoric also, the Swedish far right has won a position that allows it to foster political instability, as shown by this week’s no-confidence vote in Stockholm. Pushing back at these forces with talk of multiculturalism and inclusiveness will go only so far. A better strategy is to nail them on the democratic rule of law. That’s where the populist achilles heel is found; and it’s where the EU has tools to act, such as article 7, which can suspend EU voting rights, or European court rulings.

By this, I certainly don’t mean that the moral and legal argument for saving people fleeing war and persecution should not be made. But it may be too late now, before the May 2019 vote, to shift those parts of public opinion in Europe that have come to believe asylum is shorthand for demographic upheaval or “replacement”. Studies show European citizens overestimate the percentage of migrants in their countries (Italians believe it is three times higher than the real figure). The bare fact that migration flows have dropped steeply since 2015 does not register in perceptions. It is no coincidence that anti-immigration narratives have now spread from Europe’s hard right to its hard left – with Germany’s Sahra Wagenknecht and France’s Jean-Luc Mélenchon arguing that the arrival of migrants is a capitalist European plot to suppress workers’ wages.

At this point, to shift the argument against Europe’s extremes away from migration makes much better sense.

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Saying that the democratic rule of law is under siege holds more political potential. This is what happened on 12 September, when two-thirds of European lawmakers drew a line marking what is acceptable and what isn’t. Think of it as a case of European checks and balances at last kicking in. The resolution voted through that day is a clear indictment of everything Orbán has done to violate democratic standards, from restricting freedom of the press to undermining the electoral system. It ought to serve as a template for a wider grassroots European campaign to protect the democratic rule of law.

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Rather than lambasting Orbán for rejecting the 2015 EU refugee redistribution scheme (compulsory quotas that never translated into reality), cornering him on the dismantling of mechanisms that give citizens a proper say in democracy, and allow them to make informed decisions, is likely to be more rewarding. A better way to counter Orbán and Salvini is to focus on how they threaten what protects citizens. Populists aim to destroy the safety that comes from being able to count on an independent judge if you have been the victim of abuse; the safety that comes with getting pluralistic information, not state propaganda; the safety that comes from being confident your shop or your business won’t be choked by kleptocratic, corrupt power networks.

It helps to picture populists as a bulldozer over which a large banner reading Migrants Out has been slapped to hide the grinding wheels and huge metal shovel that are busy dismantling the democratic rule of law. It’s happened in Hungary and Poland, and it’s threatening to happen in Italy if Salvini gets his way. Ask a European citizen if they want more migrants and they may answer uneasily. Ask them if they want their government to deprive them of the tools that give people a say and the protections that come with democratic rule, and the response will be more forthright.

Rule of law – as a shield against abuse of power and corruption – should be the signature theme of next year’s election.

Choice of vocabulary matters too. Framing the debate as a battle of “progressives versus nationalists” has limits because populists will push back by equating “progressivism” with enforcing “open-border” or “anti-Christian” policies. A shrewder approach would be to cast this existential battle for Europe’s soul as “democrats versus authoritarians”. At the end of the day, our common enemy is autocracy. Arbitrary rule leaves citizens unprotected; Europe’s body of law protects them. Populists want that to come undone, so they can redraw the continent as they like. That’s where the real, immediate danger lies – not in all the fantasising that, from Brexit to Orbán, has surrounded migration.

• Natalie Nougayrède is a Guardian columnist and leader writer