Of the many myths peddled by those campaigning for Britain to leave Europe, the claim that Brexit would benefit non-European immigrant communities is among the most outrageous.

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The fact is, our communities’ legitimate concerns about the current immigration system lie at the door of the Tory government, not the EU, and the consequences of leaving would hit minority communities hardest.

The leavers’ false claim is that, if we leave the EU, there can be higher non-EU immigration while bringing down overall numbers. Let’s not be seduced by this lie from people who are anti-immigration and who have spent their lives campaigning against the interests of working people.

Let’s start with the facts. Rather than being held back, non-EU immigration is rising and is higher than EU immigration. Where there are restrictions, this government has full control and EU membership does not prevent the UK from easing rules for non-EU migrants. The Tories could do this if they wanted – but they don’t. Indeed, since 2010 they have toughened eligibility criteria for tier 1 migrants and introduced new restrictions. Where there are shortages in certain sectors, such as in our curry houses, it’s government rules that are the root of the problem.

Priti Patel, the employment minister and a prominent leave campaigner, has responsibility in this area, so she is in fact the source of the problem while claiming to be the solution.

It is simply wrong to argue that leaving Europe would strengthen our ties with the Commonwealth. The EU amplifies Britain’s ties with our historic Commonwealth partners: economically and culturally we are stronger in.

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Within the EU, Britain’s trade with the Commonwealth is flourishing as we have access to the EU’s free trade deals with 19 Commonwealth countries, and many more are in the pipeline. Our exports to those Commonwealth countries have increased by a whopping 66%.

Being in Europe has enabled us to actively support Commonwealth interests, whether overturning an EU ban on Indian mangoes or securing better EU trade terms for Pakistan after floods in 2010 and 2011. That’s why significant Commonwealth leaders have backed remain. The prime ministers of India, Narendra Modi; Australia, Malcolm Turnbull; New Zealand, John Key; and Canada, Justin Trudeau – as well as the secretary general of the Commonwealth, Patricia Scotland – have all said we are better off in, and that being in Europe strengthens our ties.

People, rightly, have concerns about the impact of immigration – and we understand – but let’s not throw the baby out with the bathwater. EU nationals, like non-EU nationals, overwhelmingly make a valuable contribution to our society, working hard, paying their way, setting up businesses and bringing up families who are as British as they are loyal to their home countries.

What we want is a system that is fair – not a free for all. That is why we must make sure that there is no undercutting of wages; that those who are here are here to work; and that communities and services are strong enough to adapt to the arrival of newcomers.

Leave campaigners want you to believe that if we leave the EU all our problems will be solved. The opposite is true. Leaving exacerbates our problems. Every single credible economic institution has forecast that our economy would take a hit. Nine out of 10 economists, the Bank of England, the International Monetary Fund and the Treasury all agree that we could return to recession, which would see people lose their jobs, push up prices, force businesses to close and put families’ financial security at risk.

Ethnic minority communities would not be immune to this pain: we would be at the sharp end of a recession. People don’t want to have to withstand another round of economic hardship just as our towns and cities are getting back on their feet after the financial crash.

And this is a double threat, because look who would be in charge if we were to leave: an emboldened Tory right wing, led by Boris Johnson, in bed with Nigel Farage. These people are the friends of discrimination and division, not progress and protection for ethnic minorities.

Do we really think that Ukip, which wants to abolish anti-discrimination law and has sought to bring xenophobia into British political debate and culture, and Johnson, who wants to scrap workers’ rights and is on record for making racist comments, will defend Britain’s minority communities? Of course not: we will be set back generations.

And let’s be honest, the dirty tricks campaigning of Tory politicians in the mayoral election has no place in our progressive society. We stood up to this divisive campaign and we should stand up to Johnson and the leave campaign’s myths on immigration too.

A vote to remain is not just a vote to keep Ukip and Johnson at bay. It’s a vote for a brighter future. Three million jobs are linked to our trade with the EU – and 800,000 will be created thanks to Europe in the coming years. Our economy will grow faster as part of the world’s largest trading bloc, which lowers prices, increases wages and opens up trading possibilities for British businesses.

Our voice in the world is louder from within the EU, whether in fighting terrorism, tackling climate change, peace-keeping, or leading on international development. Britain’s ethnic minority communities are stronger, safer and better off in Europe.

• This is from an open letter also signed by Labour black and minority ethnic MPs Rushanara Ali, Shabana Mahmood, Yasmin Qureshi, Thangam Debbonaire, Virendra Sharma, Kate Osamor, Tulip Siddiq, Chi Onwurah, David Lammy, Valerie Vaz, Khalid Mahmood