Silver City (1984)
Alamo Bay (1985)
Dheepan (2015)
Amreeka (2009)
The Emigrants (1971)
The New Land (1972) sequel to The Emigrants (1971)
In This World (2002)

The Immigrant (1917)
Utvandrarna (1971)
Nybyggarna (1972)
House of Sand and Fog (2003)
Maria Full of Grace (2004)
The Beautiful Country (2004)
The Visitor (2007)

Moscow on the Hudson (1984)
It must be 20 years since I last saw this movie but so many touching moments have stayed with me, randomly popping into mind from time to time. Definitely worth looking for.

America America (1963)
Autumn Moon (1994)
Brooklyn (2015)
The Immigrant (2013)
El Norte (1983)
The Good Lie (2014) - technically refugees
Moscow on the Hudson (1984)
Stroszek (1977)
I already have enough weird friends as it is.

Dirty Pretty Things
The Tenement > edit: that should be The Tenant. lol
Masala
Innocent (2005)
Take Out (2004)
Le Fate Ignoranti
Babette's Feast
Mississippi Masala
various Superman
Crash (2004)
There are a lot of others.
some I have not seen:
Black Legion
Fatima
La Ciudad (1998)
Ali: Fear Eats the Soul
Samba
Dheepan
There are a bunch of gay films which prominently feature immigrants. Let me know if you want a list.

The Immigrant (2014)

Thanks for your insights, paislene! I haven't seen either movie (nor even The Imitation Game) but I'm quite fond of the topic of mathematics (and the people in history who contributed to it), so those three movies are certainly on my to-watch-list.
And I hope the OP will like it too.

I have seen The Man Who Knew Infinity , and yes it is about immigration . The story of a young Indian man who became one of the Worlds legendary Mathematicians , in the face of racism in Britain . If you liked The Theory Of Everything (2014) you should enjoy this too !

• Winky's Horse (2005)
• The Visitor (2007)
• Fast Food Nation (2006)
• Die Farbe des Ozeans (2011)
• Lichter (2003)
• L'envahisseur (2011) a.k.a. The Invader
• Je vous trouve très beau (2005) a.k.a. You Are So Beautiful
• Green Card (1990)
• Le silence de Lorna (2008) a.k.a. Lorna's Silence
• Birthday Girl (2001)
• Heaven & Earth (1993)
• Desert Flower (2009)
• L'assedio (1998) a.k.a. Besieged
• Shqiptari (2010) a.k.a. Der Albaner
• Va, vis et deviens (2005) a.k.a. Live And Become
• Eize Makom Nifla (2005) a.k.a. What A Wonderful Place
• Inch'Allah dimanche (2001) - not seen myself
• Gangs Of New York (2002)
• Spanglish (2004) - not seen myself
• The Terminal (2004)
• Tombés du ciel (1993) a.k.a. Lost In Transit - not seen myself
• It's A Free World (2007)
• Le coup de sirocco (1979)
• Hop (2002)
• The Man Who Knew Infinity (2015) - possibly; not seen myself

The Terminal (2004)
is a very entertaining movie based on actual events.
I've been ionized, but I'm okay now.