Immigration won't solve the long term declining birthrates because it's a globally systemic problem. Moving people from one country to another won't actually solve the population decline globally - which is needed to support the broader global financial trade markets that have propped up the golden financial age we've experienced over the last 70 years.

"The End of the World is Just the Beginning" has been a great read so far (about halfway through it as of this comment) and the author, Peter Zeihan, dedicates a significant portion of the book to discussing the problem of population decline in-depth.