Who's to blame for the flood of children at the U.S.-Mexico border?

The political battle escalated Sunday over the recent flood of undocumented children at the U.S.-Mexico border, with Republicans blaming President Obama's failure -- or unwillingness -- to secure the border, and members of both parties faulting the GOP-led blockade of an immigration reform bill that would strengthen border security.

The administration has blamed the surge in unaccompanied minors on violence and unrest in Central America, particularly in Honduras, El Salvador, and Guatemala, where a growing number of parents have dispatched their children to the U.S., often unaccompanied, in a blind shot at a better future.

Members of both parties agreed that the children must be treated humanely and ultimately sent back to their home countries, but they differed sharply on who should be blamed for the whole mess.

Gov. Rick Perry, R-Texas, said he warned the administration as early as 2010 about the problem of minors showing up at the border, but his warnings fell on deaf ears.

"I will tell you they either are inept or don't care, and that is my position," he told ABC News. "I have to believe that when you do not respond in any way, that you are either inept, or you have some ulterior motive of which you are functioning from."

Perry charged earlier this week that the administration might even be complicit in encouraging those countries to send children north, and he doubled down on that criticism Sunday.

"I don't believe [the president] particularly cares whether or not the border of the United States is secure, and that's the reason there's been this lack of effort, this lack of focus, this lack of resources," he said. "The president has sent powerful messages time after time, by his policies, by nuances, that it is OK to come to the United States and you can come across and you'll be accepted in open arms. That is the real issue."

Rep. Raul Labrador told NBC News the American people were "frustrated" by the surge of undocumented immigrants because "they feel that this administration is doing nothing about border security."

And even Rep. Henry Cuellar, a Texas Democrat whose district lies on the border with Mexico, said the administration should have been more prepared to deal with the problem.

"With all due respect to the administration, they were one step behind," Cuellar told CNN. "They should have seen this coming a long time ago."

Homeland Security Secretary Jeh Johnson defended the administration's response Sunday on NBC, blaming a misinformation campaign from smugglers for swelling the number of undocumented children bound for America, and saying the U.S. would eventually "stem the tide."

"The criminal smuggling organizations, are putting out a lot of disinformation about supposed 'free passes' into this country that are going to expire at the end of May, at the end of June," he said.

Johnson pushed back on those who have linked the smugglers' misinformation to the administration's "deferred action" program, which was signed in 2012 and allows the children of some illegal immigrants to remain in the U.S.

"The deferred action program is for kids who came to this country seven years ago," he explained. "It's not for anyone who comes to this country today, tomorrow or yesterday. And the legislation that the Senate passed, which provides for an earned path to citizenship, is for those who were in this country in 2011. It's not for those that are coming here today."

Johnson said the U.S. had commenced deportation proceedings against all "illegal migrants, including children," but that it would look for "additional options" to process the children "consistent with our laws and our values."

Some Democrats, though, fired right back at the administration's critics. Sen. Dick Durbin, D-Ill., said Republican complaints about border security were laughable, given their opposition to last year's Senate immigration bill, which would have allocated over $40 billion in new resources to the U.S.-Mexico border.

House Republicans "had the opportunity for one solid year to call the immigration reform bill, and yet they refused to," Durbin said on CBS News' "Face the Nation." "And now they're arguing we need more enforcement at the borders and a lot of other things. When are they going to accept their responsibility to govern, to call this bipartisan bill for consideration?"

Durbin received some backup on that count from Sens. John McCAin, R-Ariz., and Lindsey Graham, R-S.C., who lamented the failure of the immigration bill during during a joint appearance on "Face the Nation" without taking a direct shot at House Republicans for spurning it.

"I can't tell you how all this breaks my heart because I still believe that comprehensive immigration reform, a major part of which was enforcing our border and reinforcing our border, and this obviously hurts that opportunity," McCain.

"The bill in the Senate doubled the size of the border patrol to 40,000," added Graham. "We can control our border. That's essential. In the bill, 90 percent operational control was required before anybody could move forward towards citizenship. There is a plan in place to control our border but it won't happen until the president signs a law."

Both McCain and Graham complained that the president's use of executive authority had poisoned the well, undermining his ability to work with congressional Republicans, but Graham reserved his sharpest words of warning for his own party.

"I don't see how you could effectively win the presidency in 2016 if you adopt self-deportation as the Republican view toward immigration," Graham said.