Your own link says there is a requirement for them to hear the asylum case, which can only happen if the refugees file as such.
If they did not file for refugee status then they are simply illegal aliens in Mexico. If they do file in Mexico and they refuse to hear their case for asylum, then they could file in the next country.
The "requirement for them to hear the asylum case" is the obligation on Mexico ONLY IF someone applies for asylum.
Asylum seekers are NOT REQUIRED to apply for asylum in the first country they get to.
> If they did not file for refugee status then they are simply illegal aliens in Mexico.
That is not correct.
Just how do you think they get on the plane without a visa or visa waiver?!?
They apply for a tourist visa, which is easier to get for Mexico than for the US. They are in Mexico, legally, then travel to the US border to apply for asylum.
Please point to the relevant section of Mexican law before you claim something is illegal.
Furthermore, for all you know, every single one of them could have answered the question "Por qué motivo viaja a México?" with "Para solicitar asilo en los EE.UU." And been allowed entry telling the truth.
By definition you can't meet the requirements for a tourist visa if you're seeking asylum as a refugee (see economic solvency requirements, job/address, etc.). And lying on federal immigration paperwork is in fact a crime in Mexico as it is in every nation on earth, so I would advise you to speak with an attorney that specializes in that area for potential punishments.