These immigrants go to western countries because of economic benefits and the quality of life there. Not because the destination country has a low birth rate.
Most of the time, the migration from under-developed countries with low education is a bad deal, economically and societal, for the receiving country. Ask Soros how much money he needs to spend to "convince" people of the contrary.
Europe waking up
The bidding war continues
Eggs in my ramen

The developed countries like Canada, Australia and New Zealand have less population as compared to area they have that's why they offer immigration opportunity. Densely populated developed countries like USA also offer immigration opportunity but it's not easy to get as compared to previous mentioned countries.
Countries like UK are now taking serious measures to stop the inflow of immigrants since they have huge number of immigrants specially from south asia.