1. my post was sarcasm
2. Cases are in Texas... Why would asylum seekers be entering the U.S. illegally through Mexico? (That's what the article said.)
Lots of folks from all over the world first get to the US through Mexico. That's what we hear all the time, and it is true. I'm not saying all asylum seekers are genuine, or even that most are -- but we have laws that allow them to enter, if genuine, and we have a process for judging them. Sometimes people actually present at the border, and say "I seek asylum"; sometimes they sneak in and then apply, and the law says that the illegal entry doesn't count against you if you apply within a year and if your claim is otherwise genuine.
SO, there's nothing impossible, or even very unusual for someone to high-tail it out of, e.g., Bosnia, Uzbekistan, China, Mauritania, Sudan, and work your way to mexico and then enter there. Again, I'm not saying that any given case is legit, or that most are, but some are, and that's exactly how it happens. They make application, it's judged by the INS (now ICE), either side can appeal, etc., etc.
Also, if you do present at the border and claim asylum, you usually get an informal interview right away. Depending on how that goes, you may be allowed to enter pending your application being processed and adjudicated. And you better believe that you will do better in that process if you have a decent lawyer. Lots of claimants have sleazy incompetent ones -- some in order to push phony claims, but some get screwed even if they have good claims.