But as the focus on military action in Syria drags former President George W. Bush’s Iraq war policy back into the spotlight, the Bush family is quietly but forcefully gearing up for another, still-developing debate: The fight on Capitol Hill over a broad overhaul of the nation’s immigration laws — a discussion critical to protecting the Bushes’ legacy on what has, for decades, been a defining issue for them.

In July, Mr. Bush, who has largely avoided the political spotlight since leaving office, attended a naturalization ceremony for newly sworn-in citizens at his presidential library in Dallas.

Former Gov. Jeb Bush of Florida, his brother, has been traveling the country delivering speeches and writing opinion pieces pegged to his recent book, “Immigration Wars,” written with Clint Bolick, which argues for change in the law.

And Jeb Bush’s two sons have been reaching out to Hispanics. George P. Bush, 37, is a founder of a political action committee — Hispanic Republicans of Texas — devoted to promoting Hispanics in Texas politics, and is running for office himself as a candidate for Texas land commissioner. Jeb Bush Jr., 29, is the founder of Sun PAC, a Florida group that recruits conservative Hispanic political candidates.

For the Bushes, immigration is deeply personal. The family chose to root its political ambitions in Texas, and Jeb Bush’s wife, Columba, is from the central Mexican town of León. The elder George Bush famously, and lovingly, once referred to his three Mexican-American grandchildren as “the little brown ones.”

Now, as the Republican Party struggles with how to attract Hispanic voters, members of the Bush dynasty seem more determined than ever to exert influence over the issue they have been helping to shape for years.

“For generations, the Bush family has been connected to Hispanics by history, geography and family, and as a result, they have a deep understanding and acute sensitivity to important cultural nuances and political issues that affect the population,” said Mark McKinnon, a Republican strategist who worked on both of George W. Bush’s presidential campaigns. “When it comes to issues affecting Hispanics, the Bush family has a strong compass.”

The family’s outreach to Hispanics is also smart politics, likely to bolster its political future in 2016 and beyond. George W. Bush won re-election to the White House in 2004 with 44 percent of the Hispanic vote, a number neither subsequent Republican presidential nominee came close to matching, and Jeb Bush is often mentioned as a likely 2016 contender in large part because of his strong relationship with Hispanic voters and support for an immigration overhaul.

Mr. McKinnon has already nicknamed George P. Bush “47.” (His uncle, of course, is “43,” and his grandfather is “41.”)

Jeb Bush speaks Spanish fluently and George W. less so, but in the words of friends, “fearlessly.”

During his 1998 re-election campaign for Texas governor, Mr. Bush made a concerted push to win El Paso, with its large Hispanic population; he won the county with just over 50 percent of the vote, and used the victory to position himself for a presidential run two years later.

“I want it to be known that a conservative candidate can carry the Hispanic vote,” he told reporters at the time.

George P. Bush said in a recent interview after a campaign event here that courting Hispanics has always been both the politically smart and morally right thing to do.

“My uncle obviously thought it was an important strategy for him, not only to win, but to expand the party, expand the base,” he said. “For my dad, it might be a little bit more personal, in the sense that he married ‘una Mexicana,’ and it certainly changes one’s perspective. But you know, being in Florida, it’s similar demographics to what you see in Texas, so it’s been important both from a political and a personal standpoint.”

In the preface to his book, Jeb Bush writes that immigration to him “means my wife and family.”

“It’s just smart marketing; it’s just smart business,” said Henry Bonilla, who is Hispanic and is a former Republican House member from Texas. “Whether it’s political business or corporate business, there are those who understand that it’s a diverse nation and it’s wise to be inclusive, and those who don’t, and it was just innately part of the Bush culture from the get-go.”