LAKE BUENA VISTA, Fla. — President Barack Obama, in a strikingly personal appeal, renewed his call for an overhaul of America’s immigration laws before a supportive Latino audience Friday. He portrayed rival Mitt Romney as an obstacle to measures that would give young illegal immigrants a path to citizenship.

“These are all our kids,” he declared.

To a standing ovation, Obama spoke of his directive last week that immigrants brought illegally to the United States as children be exempted from deportation and granted work permits if they applied.

Reflecting on his own life as the first African-American president, he said: “When I meet these young people, all throughout communities, I see myself. Who knows what they might achieve? I see my daughters, and my nieces, and my nephews.”

“That’s the promise that draws so many talented, driven people to these shores. That’s the promise that drew my own father here,” said Obama, whose father was Kenyan.

Obama spoke to the National Association of Latino Elected and Appointed Officials, gathered on the sprawling grounds of Disney World, his first speech to a Hispanic group since he announced the new deportation policy. Romney spoke to the group on Thursday, underscoring the importance of the growing Hispanic vote and the influence it could have this election year in swing states from Nevada to Colorado to Florida to Virginia.

Obama’s address also illustrated his own challenge in meeting an earlier campaign pledge.

Four years ago, before the same Latino leaders group, Obama vowed to make changing the nation’s immigration system and taking steps to legalize millions of illegal immigrants a priority he would “pursue from my very first day.”

Though hardly single-minded in their approach to politics, most Hispanics have been voting Democratic in recent elections. Obama has risked losing some support in part because Hispanics have been hard hit by the weak economy. What’s more, Latino leaders have also grown frustrated with Obama because he failed to deliver on his 2008 pledge and because his administration was deporting illegal immigrants in record numbers.

On Friday, Obama blamed the lack of broader changes on Republicans who once supported adjustments to immigration law and “have been driven away from the table by a small faction of their own party.”

He noted that Romney during the Republican primaries said he would veto legislation, known as the DREAM Act, that would give a path to citizenship to young immigrants who came to the United States illegally as children but have since attended school or served in the military.

“He has promised to veto the DREAM Act and we should take him at his word,” he said,

It was lack of action on the DREAM Act — the Development, Relief and Education of Alien Minors Act — that led him to take the administrative steps he did last week to defer deportation for some young illegal immigrants and give them work permits instead, he said.

“It falls short of where we need to be, a path to citizenship,” he said. “It’s not a permanent fix. It’s a temporary measure that lets us focus our resources wisely while offering some justice to these young people.”

The directive could benefit anywhere between 800,000 to 1.4 million immigrants. On Friday, the Department of Homeland Security said it has found 823 cases in the backlog of deportation cases that will qualify for closure under Obama’s new rule, making them eligible for work permits. The department said it expects to identify thousands more cases in the coming weeks.

Romney on Thursday assailed Obama’s action as merely a “stopgap measure” in his own speech to the association, backing off the tough anti-illegal immigration rhetoric of the Republican primaries. He promised to address illegal immigration “in a civil but resolute manner.”

Obama spoke about two hours after Sen. Marco Rubio, R-Fla., who has been promoting a plan that would have dealt with young illegal immigrants in a similar fashion to what Obama accomplished administratively.

Rubio, a possible running mate for Romney, said the issue had been politicized and neither side wanted to solve it because it was a more powerful political tool if left to fester.

“I was accused of supporting apartheid,” Rubio said. “I was accused of supporting a DREAM Act without a dream. Of course, a few months later the president takes a similar idea and implements it through executive action and now it’s the greatest idea in the world.”

Rubio said, “This issue is all about politics to some people. Not just Democrats, Republicans.” Romney has said he was studying Rubio’s proposal but has not endorsed it.

Obama tailored his usual economic message to Hispanic voters, saying he would build up-middle class opportunity for Latinos while Romney would hurt it with “top-down economics” favoring the rich.

Romney’s camp responded Friday by issuing statements from a number of prominent Hispanic leaders who said Obama’s economic policies would hurt Hispanic businesses. In one typical reaction, Republican Rep. Mario Diaz-Balart of Florida said Obama was changing the subject to avoid having to discuss the economy.

“The Hispanic community values entrepreneurship and family-owned businesses, and we deserve a leader in Washington who is dedicated to creating an environment where our values, our goals and our dreams of prosperity can become reality,” he said.