The Republican Platforms in 1956 (Eisenhower presidency; pre-switch), 1968 (Nixon presidency; middle of switch), and 1980 (Reagan presidency; post switch). I will mostly be comparing Eisenhower and Reagan. I will also be working on the Republican party, since it is its Southern Strategy that is driving this change, and the Democrats are, for the most part, following.  

Economically? Yes and no. Eisenhower advocated balancing the budget and shrinking the government, but he also pledged to not shrink any part of the security nets already in place, and promised that the first tax cuts would go to lower and middle class families, as well as smaller mom-and-pop style operations. Reagan, meanwhile, makes no mention of who would receive the tax cuts he promises, and promises to restrict the security nets those felt "worthy". He also promises to return control of welfare systems to the states, more than a few of which still had remnants of Jim Crow lingering in government.  

tdlr (economics); Both advocate balanced budget + tax cuts, but by Reagan's presidency the goal had transitioned from "well-managed government" to "small government," like we see today. Not a switch with the Democratic party.  

Foreign/Military-wise? Very little change; still fervently anti-communist, even if the battlegrounds had changed. The biggest change is the attitude toward immigrants. Compare Eisenhower's stance;

The Republican Party supports an immigration policy which is in keeping with the traditions of America in providing a haven for oppressed peoples, and which is based on equality of treatment, freedom from implications of discrimination between racial, nationality and religious groups, and flexible enough to conform to changing needs and conditions.  

and Reagan's stance;

We believe that to the fullest extent possible those immigrants should be admitted who will make a positive contribution to America and who are willing to accept the fundamental American values and way of life. At the same time, United States immigration and refugee policy must reflect the interests of our national security and economic well-being.  

Notice the difference? It goes from a haven with "freedom from implications of discrimination" to a haven for those "who will make a positive contribution to America and who are willing to accept the fundamental American values and way of life." A small change, but a very important one.  

There is also a very important change in tone between Eisenhower and Reagan. Compare;

Now, we are at peace. The Korean War has been ended. The Communist aggressors have been denied their goals.

The threat of global war has receded.

The advance of Communism has been checked, and, at key points, thrown back. The once-monolithic structure of International Communism, denied the stimulant of successive conquests, has shown hesitancy both internally and abroad  

and  

At the start of the 1980s, the United States faces the most serious challenge to its survival in the two centuries of its existence. Our ability to meet this challenge demands a foreign policy firmly rooted in principle. Our economic and social welfare in the 1980s may depend as much on our foreign and defense policy as it does on domestic policy. The Republican Party reasserts that it is the solemn purpose of our foreign policy to secure the people and free institutions of our nation against every peril; to hearten and fortify the love of freedom everywhere in the world; and to achieve a secure environment in the world in which freedom, democracy, and justice may flourish.  

One is triumphant, the other sound almost scared and alarmist. Part of this is due to the fact that Eisenhower was defending, and aiming for re-election, and Reagan attacking, and seeking to dethrone the incumbent Carter, but it, in conjunction with the shift on immigration, marks a more isolationist stance on the part of the Republican party. The idea that the world is (outside of the threat of communism) a friendly place is gone.  

The weird thing is that the USSR wasn't all that much of a threat compared to the 1950s. The looming spectre of MAD was ceasing to loom, thanks to the SALT treaties and other moves towards nuclear disarmament. A new embassy was being built in Moscow in 1979- is that the mark of an enemy? The Soviets were invading Afghanistan, true, but in 1956, the Hungarian Revolution was brutally put down, the Warsaw Pact had been established in 1955, and the first Soviet hydrogen bomb had been detonated in 1953. Where is the call to arms on the part of Eisenhower? Why is there no undercurrent of fear on the part of the 1956 Republican party?  

tdlr (foreign policy); No real shift in terms of overall policy, but massive shift in terms of tone. Also not a switch with the Democratic party.  

Environment? in 1956, the Republican party was still the party of Teddy Roosevelt. The big news in 1956 was water management, and Eisenhower promised to conserve as much as possible while still maintaining traditional state's rights, and repeatedly reaffirmed Republican support for the National Park system. Reagan, meanwhile, still supported the National Parks, and added 18 new ones during his presidency, but also added this disclaimer;

We strongly affirm that environmental protection must not become a cover for a "no-growth" policy and a shrinking economy. Our economy can continue to grow in an acceptable environment.  

Successor Republican presidents have created fewer and fewer national parks -Bush 1 founded 14, while Bush 2 founded just 7-while both Clinton and Obama founded 19.*  

tdlr (environment); Another tonal shift, in favor of business and economic concerns compared to earlier favoritism towards environmentalism. Complete 180; switches with the Democratic party.  

tdlr (overall); some switching of platforms, but it's not a complete shift.  

*This is in violation with r/askhistorian's 20 year rule, but I feel that it is necessary to show how the Republican party has switched with the Democratic party. I will remove it if anybody feels it violates the spirit as well as the letter of r/askhistorian's rules.