Contrary to the double meaning of the title of this four-part feature, globalisation’s immigration supply chain has been thoroughly documented by now, from Michael Winterbottom’s In This World (2002) to Jacques Audiard’s Dheepan (2015). But maybe, with so many of the undocumented still mistreated, discarded and forgotten on the way, it can never be re-examined enough.

It’s a little disappointing, then, that this empathetic but dogged portmanteau dithers in its first two sections, which fail to get past generic conceptions of migrants and asylum-seekers. In the first, Afghan Leila (Marjan Khaleghi), determined to rejoin her husband in London, tries to hook up with people smugglers in Tehran – but learns she is pregnant. The second sees Iranian youngster Parsa (Parsa Bahadori) and his mother stuck in safe-house limbo in Athens, trying to hide the ailing condition of a fellow immigrant from a bullying gang boss. The situations are desperate, but the drama is hesitant.

More complex scenarios with more internal conflict yield better results in Undocument’s back half. Episode three features a makeshift domestic idyll – a Polish hairdresser (Julia Krynke) and her teenage daughter living with her North African boyfriend in the UK – which is shattered when immigration officers come calling. The final section follows Arabic interpreter Ramzi (Ako Ali) at work in the immigration courts, where every nuance and omission can be life-changing.

Ali, studiously glum, shows the price Ramzi must pay for quashing his feelings to be the picture of impartiality. The courtroom TV screen, with the live feed of the Syrian mother whose case is being heard, is a simple but striking alienation effect straight out of the Iranian new-wave playbook; a fine example of the illuminating effect of influences that have travelled thousands of miles to be here.

• Undocument is released in the UK on 9 December.