They would. No country has been able to successfully turn the tide much. Singapore is a very good example due to shared culture etc. and Singapore essentially gave up and started bringing in immigrants.
The thing is as people get richer and better educated, they would want further less kids. Also, since they are rich, economic offerings may not entice them any more.
Modern Immigration Wave Brings 59 Million to U.S. | Pew Research Center
Here's the link that I want to post. The graph was from this link. Sorry to confuse you.
I was just reading that paper on textiles when I also copied, and perhaps I mixed it up a bit.