They also participated in a forum at a Border Patrol station in Weslaco.
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A top immigration official says it's unclear how family reunification will occur now that President Donald Trump ordered parents and children no longer be split.
"It's a big question. There have not been a lot of answers," said Henry Lucero, a director of field operations at Immigration and Customs Enforcement.
ICE, which detains the parents, says family reunification isn't new to the agency but the numbers are larger now. Lucero said in the majority of cases he knows during his career, the parent asks to be deported and leave the child with a caretaker, typically a relative.
Sister Norma Pimentel of Catholic Charities of the Rio Grande Valley pushed back, saying parents choose to go home without their children because it takes four months to reunite.
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Lucero disagreed, saying it "generally takes days" for ICE to reunite a willing parent with the child, who is monitored by the Health and Human Services Department.
Cruz and Cornyn said they want to keep children together with their parents.
"A strong consensus in Washington that kids need to stay with their parents," Cruz added.
The Associated Press contributed to this report.