I guess I'm lucky that my birth country's passport offered visa-free travel to 150+ countries: https://www.henleyglobal.com/passport-index/ranking
* Zweig also mentions how in the late 1930's people whose conversation was normally on a much more intellectual plane —books, opera, art, etc.— were reduced to concrete discussions centred around rumours of which countries it was still relatively easy to get visas for.
the US started restrictions (at the time by race) with the Chinese Exclusion Act of 1882.
Australia and New Zealand had their own versions of the above in place somewhere between the 1880s that and the 1920s.
Latin America also attracted a lot of immigrants so I assume not that many restrictions but I don't know what laws were in place. You'd have to check country by country.
I assume colonial powers didn't restrict their own countrymen from emigrating to their colonies in Africa or Asia but that's just a guess.
I believe the above covers most of the large scale immigration at the time. To be honest I don't know what would happen if say a random Frenchman went to Tokyo/Moscow/Beirut and tried to buy a house, get a job and marry a local but that's an interesting question.
Unfortunately the general wikipedia page isn't complete: https://en.wikipedia.org/wiki/Immigration_law
In the US at least it goes back a little bit further than WW1, to the Chinese exclusion act in the late 1800’s.
For instance, paupers were often excluded from American cities as early as the 1700's.
As for immigration quotas, early American colonies were more concerned with increasing immigration. America was big and empty. If you had money, you were welcome.
About that. If you didn't have money, you became an indentured servant. Took half your life to pay it off. Not a real citizen until then.
So, sure, you could come to America. But it wasn't all honey and roses. It was different, but not much better.
Visa, immigration status, taxes and such are quite a big factor for people to consider, particularly when they are more established in their lifes.