1. Strong majorities think that people who were brought here as children should be allowed to become citizens. Majorities say that people who came as adults who have been working and don't have a criminal record should at least be allowed to stay, and possibly to become citizens.
2. Opinion is now pretty evenly divided on whether the level of legal immigration should be increased, reduced, or kept the same.
3. Large majorities say that immigration laws should be more strongly enforced about people who are now trying to come in.
That is, give a break to people who are currently here, but try to stop further illegal immigration. The policy of the Obama administration was pretty much in line with prevailing public opinion--he supported a proposal for a "path to citizenship," established DACA, left the laws on legal immigration alone, and deported a lot of people. However, he didn't say much about the deportations. The attention came from critics, mostly on the left, but including Donald Trump in one of the debates: "President Obama has moved millions of people out. Nobody knows about it. Nobody talks about it. But under Obama, millions of people have been moved out of this country. They've been deported." Hillary Clinton said even less about them--as I recall, she just ignored Trump's statement. Why not talk about a policy that would be popular and refute Trump's claims about how we had "open borders"? There were some immediate reasons, which are discussed in this article. But I think there was also a deeper reason.
In popular moral thinking, nations are important. We have obligations to other members of our nation that we don't have to people in other nations. That raises the question of who is a member of our nation. Regardless of their views on what the right level of immigration should be, the great majority of people would agree that "we" (the current citizens) have a right to decide on their number and the conditions of joining our nation. If you asked people to give reasons for these beliefs, I think most would say that they are just common sense.
What you could call "sophisticated" thought is not satisfied with appeals to common sense--it demands justification in terms of principles. Sophisticated thought is not limited to intellectuals in a narrow sense--it also includes most journalists and politicians, and a significant number of educated people more generally. The major principle that is accepted today is human rights: people are endowed with certain inalienable rights, among which are life, liberty, and the pursuit of happiness. Nations and governments are just a means to secure these individual rights. Living where you want is a basic part of liberty and the pursuit of happiness, so on a straightforward interpretation of this principle, it's hard to justify any restrictions on immigration. At the same time, most sophisticated people also share the intuitive sense that the nation is more than just an instrument for securing individual rights. By and large, they deal with this conflict by avoiding it: Democrats denounce anti-immigrant policies but don't say much about what they think immigration policy should be, and Republicans call for the "rule of law" but don't try to give a justification for those laws.
I can't offer any direct evidence for any of this, but I think it is a way to make sense of a number of things about public opinion and politics today.