I think an originalist reading of the Fifth Amendment disagrees.
Here is Amendment V:
No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a grand jury, except in cases arising in the land or naval forces, or in the militia, when in actual service in time of war or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.
A couple of points here. First, you're forgetting the words "of law" that follow "due process". There is a law on the books. And that law is being followed. If there were no immigration laws, and immigrants were being rounded up and deported just on general principle, then you would absolutely have a Fifth Amendment claim. But that's not what is happening here. There are very specific laws on the books calling for prosecution and/or deportation of people entering our country illegally. There are also specific limitations enumerated for asylum claims. Due process.
Secondly, no unprosecuted deported person is being deprived of life, liberty, or property. They are simply being removed from the territory of the United States. So if you're going to bring up Amendment V, at least know what it actually says.