The recent endorsements of Hillary Clinton by Republican party actors are unlikely to affect how she behaves in office if she is elected. Unless, that is, this support changes the way the Republicans themselves behave.
To assess how Clinton would act as president, look at the promises she is making as a candidate. Not only do politicians try to keep their pledges. Even when they don't, their promises inform how they think about what they are doing and how to explain it to voters.
These "promises" are more than about specific policies. They also include cues on how a politician will act in office generally, whom she will listen to and, at some level, who she will be. For example, because Clinton is making a big deal of possibly being the first woman president, rather than playing this down, she'll be expected to "be" a woman in office.
Clinton's most important promise in 2016 is to be a mainstream Democrat -- a worthy successor to Barack Obama. In her primary campaign, she won a record number of endorsements for a non-incumbent from Democratic party actors across the ideological spectrum. At the party's convention, the president and Michelle Obama were the featured speakers on two of the four nights, and Vice President Joe Biden was a major speaker on another night.
Clinton refers to Obama frequently. While she has occasionally found differences with him on specific policies, such as her stated opposition to the Trans-Pacific Partnership, she has never even hinted at putting any real distance between them.
Nor does she try to separate herself from the Democrats in Congress -- not a particularly popular group. And though she doesn't have much choice about being linked to her husband's administration, she frequently brings up Bill Clinton's record (as she sees it) when she doesn't have to.
This is not all about oratory and stagecraft. Her campaign personnel are a mixture of Obama and Bill Clinton veterans, along with people from other Democratic campaigns.
So Clinton is going to be a mainstream liberal Democrat if she's elected, and that is going to be the case whether or not she receives some Republican support. She has made too many promises by this point to start recasting herself as someone primarily dedicated to uniting across the parties. Indeed, the best understanding of the Bernie Sanders insurgency wasn't that it pushed Clinton or the Democratic Party closer to him; it's that it drove her to be more specific about how she would act for the party's liberal mainstream.
So if Clinton is elected, the ball will be in the Republicans' court, not hers. Will the Clinton-endorsing Republicans or the larger group of Trump-rejecting Republicans question some of the strategies that contributed to the Trump takeover of their party? In particular, the Republicans decided back in 2009 to oppose the current Democratic president on everything, instead of being open to finding common ground. Will that change?
Listen to the anti-Trump Republicans.
If they say they were doing just fine before Trump and want to return to the status quo as soon as possible, then they will continue their strategy of blanket resistance to a Democrat in the White House. As the successor of a president who learned to deal with obstruction, Clinton will be prepared for that fight.
What if they instead blame their party for the rise of Trump? Then they will be setting the stage for change. And if that happens, expect Clinton to find herself constrained by Obama's legacy and her party to bargain with Republicans who are willing to compromise.