Refugees Welcome Here – it comes close to cliché. Well-meaning wellwishers at arrival gates brandishing signs and smiles. With every new entrance, they send up a cheer: You made it. You're here. You're welcome.

The Welcoming Party starts with just such a salutation, then proceeds to show how shallow a gesture it can be. This is, no two ways about it, an astonishing piece of theatre: a kids' show that handles immigration, both as lived experience and as bureaucratic system, with such nuance and sophistication that it teaches adults a thing or two as well.

Theatre-Rites is one of the most grown-up children's theatre companies going. Run by director Sue Buckmaster, it turned 21 this year and it's never shied away from a slippery subject, be it the banking crisis or climate change, nor scrimped on theatricality. As ever, The Welcoming Party blends puppetry, physical theatre and real-life testimony into a site-specific show.

A group of five well-wishers walks us through the 1830 Warehouse, now part of Manchester's Museum of Science and Industry, but originally the world's first railway warehouse – a place where cargo was stored, assessed and unloaded. Buckmaster's point,is that migrants are treated much the same. The floors and beams are two types of timber: African Oak from Sierra Leone and Greenheart, perhaps from Guyana. The space is rich with resonance.

It is filled with real-life stories – some familiar, some less so. Indeed, if The Welcoming Party starts with stock imagery of immigration, the sights that even kids pick up from the news, it peels back to a more complex reality.

Mohammed Sarrar, a Sudanese refugee, recounts his journey in miniature. A toy truck takes him across the Sahara, a raft stuffed with orange life jackets floats over a Cellophane sea. A puppet falls overboard and kicks against the current. He hides in train toilets and cowers in the Calais Jungle beneath a plastic sheet tent. It's a breath-taking sequence – a whole show in itself.

Yet, The Welcoming Party doesn't stop there. It reveals the journeys made by his castmates, the locals braced to welcome him at the start. Amed Hashimi takes us through his migratory childhood, from Baghdad to London via Norway and Iran, with each new home housed in a box file like a pop-up puppet theatre; each a memory filed away. Emmanuela Yogolelo conjures her native Congo through songs and sayings, a culture she carries with her, and Michal Keyamo strains to recall her scant memories of arriving from Nigeria at five, a puppet in her mother's arms. There are more migrants here than first meets the eye.

Why, then, is the system so set against new arrivals? As Mohammed navigates a stern and impersonal system, caged for questioning and processed like a package, the show stops feeling like a party. One by one, we all fall under suspicion, and the show sweeps us into a bureaucratic black hole. We're forced to fill in nonsensical forms, a welcome shot of activity, to prove our right to remain. We're shunted this way and that, held here, sent there – a brilliant encapsulation of an arbitrary and authoritarian system. By submitting us to it, adults as well as kids, The Welcoming Party lets us see the story from inside. That it's so simply told makes it all the more pressing: a checkpoint for something straightforward that, if unchecked, risks spiralling out of control. This is vital – in every sense of the word.

The Welcoming Party runs at the Manchester Museum of Science and Industry until 16 July.