Well, the most natural thing would be for the new agreement with Mexico to work the same way as the existing agreement with Canada. According to Just Security:
The U.S.-Canada agreement requires each country to return an asylum seeker to the other country for her/his claim to be adjudicated if the person was “physically present [in the other country] immediately prior to making a refugee status claim.”
I.e. people who travel through Mexico to make an asylum claim in the US will be deported to Mexico to make their asylum claims there.