Well, Republicans have been moving to the left for a while too...

GWB ran on the idea of "compassionate conservatism" which was another way of saying that the Federal government should spend more money on social welfare programs. For GWB, it meant ensuring the support of a wide variety of faith-based organizations who saw it as an opportunity to get some free money, and when it worked to help get GWB two terms, the GOP realized it was a winning strategy. It helped counteract some of the appropriations championed by Democrats that resulted in lots of loyal voters.

Similarly, post-911 the neoconservative faction of the GOP took a strong hold on power. The older generation of Republicans, many of whom had served in combat, were far more weary of costly foreign wars, but the young generation, most of whom had not served, were eager to seize upon the opportunities left around the globe after the fall of the Soviet Union.

The invasion of Iraq was neoconservative: It was sold on the basis of its alleged humanitarian objective (freeing women who were oppressed under Saddam) but also on the basis of the grand existential threat of terrorism. Americans had to be sold on the idea that the war made sense, and GWB and his neoconservative team created a villain out of Saddam, Hans Blix the UN weapons inspector, the European countries that urged restraint, etc.

Of course, this was all so that the US could invade a country that had once been part of a proxy war with the USSR and was now essentially available for the taking. We've seen US dependence on Israel decrease post Iraq invasion, as we now have our own territory for bases and our own projection of force in the region. The current situation with Syria is Russia's latent attempt to retaliate and not lose its last foothold in the middle east. The Iran deal was (seemingly unsuccessfully) intended to be a carrot preventing Iran from aligning with Russia, etc.

Neoconservatism has been described as "liberals with guns" because there is an element of a moralistic crusade behind the war, and the war often does not involve direct threats to national security. Post Saddam, the US doled out the Iraqi oil industry contracts to American and coalition firms, effectively redistributing Iraq's trillions in oil wealth substantially, cutting out French and German companies that had won these lucrative contracts before.

In spite of campaigning against Bush's wars, Obama has generally kept the same ideological approach. For all the ranting about Obamacare, the GOP largely supported it, mainly because there are enough firms in GOP districts to result in the required number of votes.

At this point, the main reason the GOP would want to change Obamacare is to make it more lucrative for specific firms, not because of any sort of philosophical disagreement with the program.

So both parties are converging on what I'd call corporatist neoconservative centrism. It generally preserves the status quo, the broad policy strokes are bold plays to enrich massive industries through public/private partnerships (middle eastern oil, US healthcare, US housing, US financial services, etc). Foreign policy is heavily propaganda-based and heavily moralistic in its tone, foreign leaders are subject to ad hominem attacks, and existential evils are spoken about with the wide-eyed seriousness of children discussing fairytales.

When you think about, say, Jeb Bush's platform and HRC's, there is little difference when you filter for issues that would ever actually come to a vote. Notable in the leaked emails was the HRC team scrambling early on in response to a speech by Jeb Bush which they described as "HRC agrees with all of this". They were fearful that his messaging approach was better than what they'd come up with.

The biggest beneficiaries of the idea that HRC is "extremely liberal" or that Jeb is "extremely conservative" are the candidates themselves. They create the illusion of difference were little exists. Many of the other leaked HRC emails show that she holds or recently held many socially conservative views to go along with her free-trade pro-wall-street economic views. These things aren't a surprise, but they do provide actual evidence of the profound similarity between the two parties.