As Hillary Clinton and Jeb Bush formalize their presidential candidacies over the next few days, both face the complex challenge of adapting their family legacies to their parties' new dynamics.

Compared with Bill Clinton's era, the Democratic Party today is more culturally liberal and economically populist. Compared with George W. Bush's era, the GOP is more dogmatically committed to shrinking government. These changes have presented Hillary Clinton and Jeb Bush with the common puzzle of appealing to parties that have grown more ideologically militant without renouncing the policy agendas and political strategies associated with their family names—agendas and strategies that often defied each party's traditional orthodoxy.

2016 presidential candidates Hillary Clinton and Jeb Bush. (Scott Olson and Richard Ellis/Getty Images )So far, this test has stumped Bush more than Clinton—as underscored by the campaign-staff reshuffle the former Florida governor announced this week, just before he is due to officially declare his candidacy next Monday. His lackluster first months exploring the race have been dominated by questions of where he would extend the policies of his brother, George W. Bush. That has exposed Jeb Bush to darts from both ideological conservatives and party pragmatists most concerned about finding a candidate who can win.

The pragmatists were dismayed by Bush's recent struggles to explain what he would have done differently than his brother in Iraq. That ordeal left Republicans fearing that if the party nominates Bush, Democrats would find it too easy to convert the campaign into a referendum on returning to the policies of the last Bush administration.

Bush's problems with the Right are rooted in two other elements of his brother's legacy. Though staunchly conservative on most issues, George W. Bush backed a pathway to citizenship for undocumented immigrants and an aggressive role for Washington in education reform. Politically, each idea was intended to court voters beyond the GOP base.

(RELATED: Why Jeb Bush Can't Spend His Way to the Win

Conservatives chafed against those policies during George W. Bush's presidency and, after he left office, successfully eroded support in the party for both ideas. Jeb Bush, though, threatens that victory. The younger Bush has said he would accept either a pathway to citizenship or permanent legal status for the undocumented, and he has defended the Common Core curriculum reform (while rejecting President Obama's effort to advance it through federal policy).

Can Bush win these arguments in the GOP? Despite loud resistance from prominent conservatives, "Jeb Bush's view on immigration is "¦ more acceptable to Republican primary voters than most people assume," notes Peter Wehner, senior fellow at the Ethics and Public Policy Center. While many Republican voters view immigration skeptically, in the latest Pew Research Center survey, nearly three-fifths of the party (including GOP-leaning independents) said the undocumented should be allowed to remain legally inside the country. That number reached nearly two-thirds among the college-educated Republicans who are Bush's natural constituency.

Bush's continued support for Common Core may be a tougher sell with Republican voters. But it's probably less important for Bush to win the specific debates over immigration and education than to subsume both issues beneath bold new domestic and foreign policy ideas that excite GOP voters. So far he hasn't done that. Unless Bush can shift his campaign's focus toward the country's future, he's likely to remain stuck in debates over his party's past. And driving in reverse is no way to win a race.

(RELATED: Jeb Bush to National Review: 'I Love You,' But 'You're Wrong on Immigration.')

Hillary Clinton, who kicks her campaign into higher gear with a major address on Saturday, hasn't faced nearly as much pressure yet within her party but could eventually confront her own legacy trap. Her announced rivals, Sen. Bernie Sanders of Vermont and former Maryland Gov. Martin O'Malley, have denounced free-trade and financial-deregulation policies that Bill Clinton pursued. Other Democrats worry about the Clinton family heritage of ethical controversy. On both fronts, Hillary Clinton's challenge will be less to defend that record than to transcend it.

On social issues such as gay marriage, Hillary Clinton has already embraced the Democrats' more liberal consensus. But for all her fiery economic rhetoric, it's not clear where she will land between her husband's business-friendly, deficit-conscious centrism and her party's rising populist current. It seems inevitable that Clinton, like Obama, will propose more responsibility for Washington than her husband envisioned when he declared, "The era of big government is over." Less certain is whether she will challenge her party to simultaneously reform government, as Bill Clinton did when he restructured welfare and balanced the federal budget. If she embraces reform (for instance, by streamlining entitlements for seniors to fund investments in kids), she risks antagonizing the Left; if she doesn't, she risks helping Republicans tag her as a return to uncontrolled big-government liberalism.

Like Jeb Bush, Hillary Clinton has to convince Americans that she is offering not just a dusted-off collection of ideas stored in the family attic but an agenda attuned to the challenges people face today. Their famous families make almost everything else about running for president easier, but no contenders may find that bar harder to clear than the candidates named Clinton and Bush.

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