It isn't clear to me why "due process" requires allowing someone who entered illegally to remain in the US while their often bogus asylum claims are adjudicated. Those applying for refugee/asylum status BEFORE entering the US aren't allowed in until after their claim is found valid.
And, as a point of reference, some countries (looking at you Australia) invalidate all future asylum claims if the applicant enters the country illegally.
Somewhere, there needs to be a point of balance between a humanitarian refugee policy and the current ploy of simply making the claim being a "get out of jail free" card for border crashers.
I don't believe that "due process" is what requires that, rather than is what the current federal statutes on the matter provide for so as that those legitimately seeking asylum don't get slaughtered back home. What due process requires is that cases and controversies be heard by a neutral and detached judicial officer as laid out in Article III.