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.
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.