The case of the Haitian migration to Chile would make a great case study. Nevertheless, it is not clear to me whether it would fit into you category of 'non-western'. In Southeast Asia, Malaysia could potentially be a good case study too since its flux of immigrants has increased in recent decades.

I think that you may like to the UNDP reports on the two Latin American refugee crisis of Venezuela and Central America. The recipient countries fit well into the label ”non Western ”.

Following up on Dr. Andrade's comment, I believe Peru is tracking--or trying to--the number of Venezuelan immigrants in the last two years or so, which is well lover 100,000. Also Costa Rica should have data regarding Nicaraguans who have fled there since the Ortega regime began cracking down on protesters last April as well as Hondurans and Salvadorans who are fleeing gang violence. The latter numbers will be small, especially relative to the Nicaraguans.

The UN Global on Migration held a conference recent. Here’s a link to the UN Global Pact websit. Look thro conference papers. https://refugeesmigrants.un.org/migration-compact