Well, syntactically you are correct. However, according to the 14th Ammendment, as U.S. citizens, they (the babies) become eligible to sponsor for legal immigration most of their relatives, including their illegal-alien mothers, when they turn 21 years of age, thus becoming the U.S. "anchor" for an extended immigrant family.
Until 1965, people admitted to the country legally had no automatic right to bring their families. After 1965, minor children of immigrants had the right to sponsor immigrants. This is what produced the chain migration that has emptied entire Mexican villages.