Demand for housing in Canada has in part been driven by foreign non-residents real estate speculators who are looking to park their money in Canada, and other places. New Zealand banned all non-resident foreigners from buying property there [1,2].

I would assume that if a population remains stable as a result of migration, that as many people are leaving houses as entering them, and there shouldn't be substantial pressure on housing prices. I don't have specific data for this.

[1] https://www.seattletimes.com/business/real-estate/vancouver-...

[2] https://www.bbc.com/news/world-asia-45199034