Walk into the council chamber at Battersea Arts Centre in south London and it feels as if you have stumbled into a very strange art gallery or museum. Beautifully lit and displayed on white plinths are photographs and everyday objects, none of any monetary value, the oddest of which is probably a fake plastic leg. Museums display what is valued within a culture, but BAC’s collection of objects are of personal importance only. It reminds us that a culture is made up of individual stories about who we are and how we understand ourselves and value each other.

In 2013, BAC created London Stories, a series of intimate encounters between audiences and ordinary Londoners who told us about their lives. The process helped us to connect with a stranger. Part of its joy was that it showed how liberating it can be to tell a story – and how the listener can be changed, too, just as in Clare Patey’s London international festival of theatre project A Mile in My Shoes.

Facebook Twitter Pinterest Storyteller Lemmar narrates in London Stories. Photograph: Joyce Nicholls

BAC has now returned to the format, but this time with an emphasis on migrants’ stories. In one of BAC’s artists’ bedrooms, Debbie sits perched on the edge of a bed and tells me about her long search for her Kuwaiti mother who arrived pregnant in London and gave her up for adoption. In the lengthening shadows of another space, Eiad describes his journey from Syria to London via leaky wooden boats and people smugglers. In the kitchen, surrounded by the paraphernalia of everyday life, Antonis tells of growing up gay in Cyprus and discovering acceptance and love in London.

Near the top of the building there is an encounter on film with Lily, who survived Auschwitz. Around her neck she wears a gold pendant that her brother hid in her shoe: “The only bit of gold that went into and out of Auschwitz still owned by the same person.”

These stories take on an extra charge when you then visit the museum at the end of the evening. In most museums you see an object and then read the interpretation of it. But BAC has inverted the model: you will already know the story behind some of the objects and photographs displayed by the time you stumble across them. That’s important, because knowing the story creates empathy and context for the objects. The meaning is magnified.

The whole experience has been carefully put together. Newspaper headlines pasted to the walls remind us that hostile attitudes towards immigrants are nothing new. The cards scattered around the building tell the history of migration to the city from 1500 onwards, recalling a time when some people were so concerned by the 3,000 foreigners in the city that it was said “Tottenham has turned French”. In the 21st century, London rings to the vibrant sounds of countless languages: the city has survived and benefited from wave after wave of migration from across the world.

Facebook Twitter Pinterest London Stories at Battersea Arts Centre.

Photograph: Joyce Nicholls

At a lunch with friends recently, talk turned to who we were and how we described ourselves. All of the 14 of us around the table were proud to call ourselves Londoners, and yet our grandparents or great-grandparents or great-great-grandparents had all come from elsewhere: Scotland, Wales, Ireland, Russia, Hong Kong, Lithuania, Iran, Germany, Poland. Most of us are migrants in one way or another, and London Stories – which will eventually also be available in an online format – is an exquisitely judged reminder, at a time when our country seems intent on pulling up the drawbridges, that we are all buffeted by the tides of history and all looking for a place that opens it arms to us and which we can call home.