Well, countries that have high automation will have higher unemployment, so it makes sense for them to have as little immigration as possible. They will need less population, not more. They may even end up with emigration as people may go back to their countries if they are unemployed. Probably countries will give incentives to encourage emigration (or they won't offer them incentives to stay), if they reach high levels of unemployment as automation rises
Automation and lower birth rates go hand in hand. Maybe by accident but it happens, and I see it as a potential (partial) solution to the future unemployment. Emigration may be another potential solution
tl;dr:
Automation = Less jobs. Low birth rates = Less people. Emigration = Less people
Less people + less jobs + emigration, should balance things out a bit