Currently the left is using a bunch of entitlement programs to entice millions of immigrants to flood into the system, and overwhelm the voting demographics to become more leftist. But eventually these entitlement programs will go bankrupt, and the only people who will want to come here will be people who are motivated and ambitious and entrepreneurial. Also, culture is a two way street. We saw this with black people, when the left thought that mixing black and white culture would cause the country to become more leftist, they strongly encouraged it. But it turned out that a lot of black people liked having freedom and opportunity, liked being entrepreneurs, liked family, work ethic, and community, and didn't like entitlements and large government. As soon as this happened, the left started trying to do everything they could to isolate blacks from white culture.
I predict that the same is going to eventually happen with immigrants. They will do everything possible to isolate the rest of the world from western culture, they will encourage us to lock out anybody else, and encourage them to not come to the USA and be "oppressed" by white people.
The world is a big place. The pattern is the same in every single last country. The Left goes out of their way to find the most radical, violent sects of every religion then imports it in the name of care and fairness. Demonizing the majority religion in public mass media while simultaneously demonizing minority religions on obscure right-wing forums creates the illusion we are expected to swallow
Except the left is not rational or logical.
Waiting until the system is bankrupt would be quite damaging to the country
This is because the leftist position is not pro-immigration for the sake of itself. The left and the right aren't defined by the policies they believe in, but rather in their respective worldviews. The left gravitates towards a stateless (nationless) classless moneyless society, while the right considers those fundamental. This is, of course, assuming a lot, since "the left" and "the right" are such broad labels, they're often almost meaningless.
It's the world elite and government that will determine when a flip on immigration happens if it ever does.
The two main reasons for immigration in western countries was to combat lower birth rates and to reduce the influence of traditional western values in western countries so the countries become easier to control by the elite. An added side benefit is being able to use immigrants to redistribute wealth from the poor to the elite.
I think you may see a flip on immigration if a new culture becomes prominent. Like in France how Macron appears to be slightly shifting on bringing Muslims in. This is because the Muslims turned out to be harder to control by the western elites than anticipated. The last thing western elites want is for a group like Muslims to seize control of western countries and usurp the western elites. Immigration will remain the main policy so much as it serves the elite which depends on how well they can maintain their control and enforce their ideologies on new immigrants. I think some cracks might start to show in immigration in the next half century. A lot of immigrants are just imo pretending to go along with the mainstream narrative because they are a minority but once they gain more power they'll start pushing their more traditional cultural beliefs which may become a problem for western elites who may then try to shift tactics with regards to immigration.