Republican presidential candidates Marco Rubio, Donald Trump and Ted Cruz stand up for the national anthem during a presidential debate in Coral Gables, Fla., on Thursday. (Pedro Portal/the Miami Herald via AP)

Sens. Ted Cruz and Marco Rubio on Sunday partly blamed Donald Trump and his rhetoric for the increase in violence at the Republican front-runner's rallies.

Cruz, appearing on NBC's "Meet the Press," said he was troubled by Trump's rallies.

"In any campaign, the responsibility starts at the top," Cruz added on ABC's "This Week." "And it is not beneficial when you have a candidate like Donald Trump, who's telling his protesters, 'Punch that guy in the face.' ... I don't think you should be encouraging people to violence."

On "Meet the Press," Trump denied any responsibility for the protests. Instead, he said he deserves credit for canceling a rally in Chicago last week over security fears involving protesters. "A lot of people have praised me for canceling the one rally. We had 25,000 people coming, we got a lot of them not to come through notice. ... I don't accept responsibility, I do not condone violence in any shape."

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During interviews on Sunday, Rubio doubled down on his criticism of Trump, whose rhetoric he said Saturday reminded him of "third-world strongmen." On CNN's "State of the Union," Rubio said images of the violence at the canceled Trump rally in Chicago "look[ed] like something out of the third world."

"I think all the gates of civility have blown apart," the senator said on CNN. He added: "This notion that a president can say anything they want, or even a presidential candidate can say anything they want, whatever comes to mind — it's just not true, and it's reckless and dangerous."