2 Be --
Let me assure you -- you need not educate me about refugee resettlement.  I worked for several years with a resettlement agency.  Some of my clients are refugees who file for permanent residence.  I am intimately familiar with the process and the problems caused by the EO.  
Also, a federal judge recently ordered a stay on parts of that Order, and the 5th Circuit refused to lift that stay.  
Asylum is an immigration benefit.  I regularly work with people seeking asylum.  Many people who are considered by some to be coming for purely economic reasons are coming here because they fear some sort of persecution.  Unfortunately,  it's very hard to get asylum, so many people do not, although I think many of them should.  
People in detention centers can certainly file asylum claims.  The problem is that with a difficulty to access legal help, their asylum claims stand little chance.