Alien

Nope, an immigrant is someone who comes with permission. Now, if you object to the term “alien,” I think “invader” would do just as well.

I vote for "Unwanted Nomads".

Are “Border Raiders” or “Paperless Pirates” appropriate?

Why not? Today, apparently, anything goes.
Emotional Support Alligator? Go for it!
Call yourself a "non- binary hibiscus"? Okily dokily!

I submit the following:

"Future Democrat Subjects"
"Non- Indigenous Persons of Questionable Origins"

Those both fit.

There are lots of different types of aliens

Legal Alien - someone here with appropriate permission
Illegal Alien - someone here without permission